LLJ< OU 166116
73
^ CQ
American State Government
AMERICAN STATE
GOVERNMENT
By
W. BROOKE GRAVES
CHIEF OF THE STATE LAW SECTION
LEGISLATIVE REFERENCE SERVICE
LIBRARY OF CONGRESS
wrTHIRD EDITION
D. G. HEATH AND COMPANY : Boston
THE THEORY OF
RELATIVITY
BY
C. M0LLER
PROFESSOR OF MATHEMATICAL PHYSICS
IN THE UNIVERSITY OF COPENHAGEN
OXFORD
AT THE CLARENDON PRESS
Oxford University Press, Amen House, London E.C. 4
GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON
BOMBAY CALCUTTA MADRAS KARACHI CAPE TOWN 1DADAN
Geoffrey Cumberlege, Publisher to the University
FIRST EDITION 1952
REPRINTED LITHOGRAPHICALLY IN GREAT BRITAIN
AT THE UNIVERSITY PRESS, OXFORD
FROM CORRECTED SHEETS OF THE FIRST EDITION
1955
PREFACE
THE present monograph is a somewhat extended version of a course of
lectures which I have given at the University of Copenhagen during the
last twenty years. Consequently, it is primarily a textbook for students
in physics whose mathematical and physical training does not go
beyond the methods of non-relativistic mechanics and electrodynamics.
The intention has been to give an account of what may be called the
classical theory of relativity in which all quantum effects are disregarded.
In view of the paramofintx importance of quantum phenomena in
i *^ ?* f^
modern physics, the limitation of the subject to classical phenomena
might be considered a serious defect of the book. However, there are
several important reasons for such a limitation of the subject. At
present, a complete self-consistent reiativistic quantum theory does
not exist. Moreover, the classical theory of relativity, which by itself
gives an admirably precise description of a very extended field of
physical phenomena, must be the starting-point for the future develop-
ment of a consistent reiativistic quantum theory. For a student and
research worker in this field an intimate aquaintance with the prin-
ciples and methods of the classical theory of relativity is, therefore,
just as indispensable as is the knowledge of the methods of Newtonian
mechanics for a real understanding of ordinary quantum mechanics.
Apart from this, the classical theory of relativity is one of the most
fascinating and beautiful parts of theoretical physics on account of its
inner consistency and the simplicity and generality of its basic assump-
tions.
The presentation of the subject in the present volume differs some-
what from the usual one in that the four-dimensional formulation of
the theory plays a less dominant role than in most of the current text-
books. Certainly the four-dimensional representation, which is based
on the symmetry between the space and time variables revealed by the
discovery of the Lorentz transformation, is the most elegant way of
expressing the principle of relativity in mathematical language^ and it
has been of the utmost importance for the rapid development, particu-
larly of the general theory of relativity. In the early books on relativity
it was, therefore, quite natural to emphasize as strongly as possible
this newly discovered similarity between the space and time variables.
However, in a textbook of today I think it is useful to stress again the
fundamental physical difference between space and time, which was
vi PREFACE
somewhat concealed by the purely formal four-dimensional representa-
tion.
In the first three chapters we have, therefore, avoided any refer-
ence to the four-dimensional picture, and the kinematics and point
mechanics of the special theory of relativity are fully developed by means
of the usual three-dimensional vector calculus. But in the following
chapters also, where the elegant methods of the four-dimensional tensor
calculus are developed and applied, a three-dimensional formulation,
which gives a better insight into the physical meaning of the theory, is
frequently given. As an example, T shall mention the treatment in
1 10 and 111 of a freely falling particle in a given gravitational field.
The motion of the particle is, of course, completely described by the
statement that the time track of the particle in 4-space is a geodesic
line, but this quasi-geometrical description does not convey a physical
understanding of the phenomenon. In the three-dimensional physical
space, however, the motion of the particle can be described by an
equation of motion of the same type as that for a particle subject to an
arbitrary force in a system of inertia, the only difference being that the
geometry in the physical space is in general non-Euclidean. In this
way we obtain definite expressions for the gravitational force on the
particle as well as for the mass, momentum, and total energy of a
particle moving with arbitrary velocity in a given gravitational field.
The three-dimensional point of view thus leads to a reintroduction of
dynamical concepts into the gravitational theory, which, I believe,
makes it easier for the student fully to grasp the physical content of
the general theory of relativity.
Since a real understanding of a physical theory is possible only
through an intimate knowledge of its predecessors, the whole of
Chapter I has been devoted to an historical survey of the difficulties of
the non-relativistic theories. Many students who intend to specialize
in experimental physics may feel that the time and effort which are
needed to learn the methods of the general tensor calculus are out of
proportion to the use they can make of this formalism in their future
work. Such readers will find the main results of the special theory of
relativity in the first thirty -eight sections. They will also be able to
read Chapter VIII and in this way obtain an insight into the ideas
underlying Einstein's general theory of relativity without spending any
time on the laborious task of learning its special mathematical methods.
We have included only those developments of the theory of relativity
which can be regarded as safely established, the various attempts at
PREFACE vii
constructing a unified theory of gravitation and electromagnetism
falling outside the scope of the present book. Also the cosmological
problems have only been touched upon, since these problems have been
extensively treated by Tolman in this series of monographs. Within
these restrictions it is hoped, however, that the reader will find a fairly
complete and well-rounded account of one of the most beautiful chap-
ters in the history of science, which for the main part was written by
a single man, Albert Einstein.
On completion of this work I gratefully acknowledge the help and
advice which I have received from many quarters. First of all I want
to express my deep gratitude to Professor Niels Bohr for his kind
interest in my work during all these years and for the constant inspira-
tion derived from many discussions and conversations at his institute.
My thanks are due to Professor N. F. Mott and Professor I. N. Srieddon
for reading the manuscript and eliminating the worst danicisms. I also
wish to thank the staff of the Clarendon Press for their friendly co-
operation.
I am indebted to Dr. W. Kohri and Dr. W. J. Swiatecki for many
suggestions which have considerably improved the text and, in particular,
to mag. scient. J. Lindhard who has been of great help in checking
all the equations and reading the proofs. Finally, I am grateful to
Miss S. Hellmann for her untiring assistance in the preparation of the
manuscript and the proof-reading.
C. M.
COPENHAGEN
November 1951
CONTENTS
CHAPTER I. THE FOUNDATIONS OF THE SPECIAL THEORY OF
RELATIVITY. HISTORICAL SURVEY .... 1
1. The relativity principle of mechanics. The Galilean transformation 1
2. The special principle of relativity . . . .4
3. In variance of the phase of a plane wave . . . .6
4. Transformation of the characteristics of a plane wave. , . 8
5. The Doppler effect ... 8
6. The velocity of light in vacuo . 10
7. The velocity of light in refractive media _ . .15
8. Hoek's and Fizeau's experiments . . . 17
9. Lorentz's theory of electrons .... 20
10. Agreement between the ether theory and the principle of relativity
as regards all effects of the first order. Format's principle 22
ll. The aberration of light ... . 25
12. Michelson's experiment . ... 26
%8 13. The contraction hypothesis ... 28
14. Validity of the principle of relativity for all physical phenomena . 29
CHAPTER II. RELATIVTSTIC KINEMATICS . .31
15. Simultaneity of events . . .31
16. The relativity of simultaneity . . .33
17. The special Lorentz transformation . 36
f 18. The most general Lorentz transformation 41
I 19. Contraction of bodies in motion . . 44
20. The retardation of moving clocks. The clock paradox . ' 48
21. Transformation of particle velocities . . 51
22. Successive Lorentz transformations. The Thomas precession . 53
23. Transformation of the characteristics of a wave according to the
theory of relativity . . . . . .56
24. The ray velocity in moving bodies . . . .58
t 25. The Doppler effect, the aberration of light, and the dragging pheno-
menon according to the theory of relativity . . .62
CHAPTER III. RELATIVISTIC MECHANICS . .67
26. Momentum and mass of a particle . . . .67
27. Force, work, kinetic energy .... 70
28. Transformation equations for momentum energy and force . 71
29. Hyperbolic motion. Motion of an electrically charged particle in a
constant magnetic field . . . . .74
30. Equivalence of energy and mass . . . . .77
CONTENTS ix
31. Inelastic collisions. Mass of a closed system of particles . . 82
32. Experimental yenficatioii of relativistic mechanics . . 85
CHAPTER IV. FOUR -DIMENSIONAL FORMULATION OF THE
THEORY OF RELATIVITY: TENSOR CALCULUS . . 92
33. Four -dimensional representation of the Lorontz transformation . 92
34. Lorentz contraction and retardation of moving clocks in four-
dimensional representation . . . . .96
35. Covanance of the laws of nature in four-dimensional formulation . 97
36. The four-dimensional line element or interval. Four-vectors . 99
37. Four-velocity and acceleration. Wave-number vector. Four-ray
velocity . ... 101
^38. Four-momentum. Four -force. Fundamental equations of point
/ mechanics in four -dimensional vector form . . .104
x 39. Tensors of rank 2 . . . . . .108
's 40. Angular momentum and moment of force in four-dimensional
representation . . . . . . .110
'41, Tensors of arbitrary rank . . .111
42. Pseudo-tensors ... . .112
43. The Levi-Civita symbol . .113
44. Dual tensors . . . . . . .114
45. Infinitesimal Lorentz transformations. Lorentz transformations
without rotation . . . . . .117
46. Successive Lorentz transformations . . . .118
47. Successive rest systems of a particle in arbitrary rectilinear motion
and in constant circular motion . . . .121
48. Tensor and pseudo -tensor fields. Tensor analysis . .125
49. Gauss's theorem in four-dimensional space . . .128
50. The fundamental equations of mechanics for incoherent matter . 130
51. The kinetic energy-momentum tensor . . .136
RAFTER V. ELECTRODYNAMICS IN THE VACUUM . . 139
52. The fundamental equations of electrodynamics in the vacuum.
Four-current density for electric charge . . .139
53. Covanance of the fundamental equations of electrodynamics
under Lorentz transformations. The electromagnetic field
tensor ........ 141
54. The four-potential. Gauge transformation . . . 143
55. Four-dimensional integral representation of the four-potential . 144
56. Retarded potentials. Lienard-Wiechert's potentials for point charges 147
57. The field of a uniformly moving point charge . . .151
58. The electromagnetic forces acting on charged matter . .154
59. Variational principle of electrodynamics . . .157
60. The electromagnetic energy -momentum tensor . 159
61. The total energy-momentum tensor . . .161
x CONTENTS
CHAPTER VI. GENERAL CLOSED SYSTEMS. MECHANICS OF
ELASTIC CONTINUA. FIELD THEORY . . . .163
62. Definition of a closed system ..... 163
63. Four -momentum arid angular momentum four-tensor for a closed
system . . . . . . .166
64. Centre of mass . . 170
65. The fundamental equations of mechanics in elastic contmua . 173
66. Transformation of elastic stress, momentum density, and energy
density . . . . . .179
67. Perfect fluids . . 181
68. Scalar meson fields. General field theory . .184
CHAPTER VII. NON-CLOSED SYSTEMS. ELECTRODYNAMICS IN
DIELECTRIC, 1 AND PARAMAGNETIC 1 SUBSTANCES. THERMO-
DYNAMICS ... .... 188
69. General properties of non-closed systems . . .188
70. Static non-closed systems . . .191
71. Electrostatic systems. Classical models of the electron . .192
72. The fundamental equations of electrodynamics in stationary
matter. . . . . . . .195
73. Mmkowski's field equations m uniformly moving bodies . .196
74. The constitutive equations in four-dimensional language. Boundary
conditions . . . . . .201
75. Electromagnetic energy -moment urn tensor arid four-force density 202
76. The propagation velocity of the energy of a light wave in a moving
refractive body ..... 206
77. The laws of thermodynamics in stationary matter . .211
78. Transformation properties of the thermodynamical quantities . 212
79. Four/dimensional formulation of the laws of thermodynamics 214
80. Ideal monatomic gases . . . . .215
81. Black-body radiation . .... 216
CHAPTER VIII. THE FOUNDATIONS OF THE GENERAL THEORY
OF RELATIVITY . . . . . . .218
*82. The general principle of relativity . . . 218
^ 83. The principle of equivalence ..... 220
84. Uniformly rotating systems of coordinates. Space and time in the
general theory of relativity ..... 222
85. Non-Euclidean geometry. The metric tensor . . 226
86. Geodesic lines . . . . . . .228
87. Determination of the metric tensor by direct measurements.
Geometry in n-dimensional space . . . .231
88. General accelerated systems of reference. The most general ad-
missible space -time transformations .... 233
CONTENTS xi
89. Space and time measurements in an arbitrary system of reference.
Experimental determination of the functions g lk . . 237
90. The spatial geometry in the rotating system of reference . 240
91. The time tracks of free particles and light rays . . . 244
92. The dynamical gravitational potentials .... 245
93. The rate of a moving standard clock in a gravitational field . 247
94. Transformation of coordinates inside a fixed system of reference . 248
95. Further simple examples of accelerated systems of reference . 250
96. Rigid systems of reference with an arbitrary motion of the origin 253
97. Rigid frames of reference moving in the direction of the X-axis . 255
98. The clock paradox . . . . . .258
i CHAPTER IX. PERMANENT GRAVITATIONAL FIELDS. TENSOR
CALCULUS IN A GENERAL RIEMANNIAN SPACE . . 264
99. Four-dimensional formulation of the general principle of relativity
and of the principle of equivalence . . . 264
100. Contra variant and covanant components of a four-vector 266
101. Tensor algebra . . . . . . .269
102. Pseudo -tensors. Dual tensors . . 270
103. Geodesic lines. Christoffel's formulae . . . 272
104. Local systems of inertia . . . . .274
105. Parallel displacement of vectors . . .276
106. Tensor analysis. Co variant differentiation . .279
107. The curvature tensor ...... 284
108. The contracted forms of the curvature tensor . 286
CHAPTER X. THE INFLUENCE OF GRAVITATIONAL FIELDS ON
PHYSICAL PHENOMENA . . . . . .288
109. Mechanics of free particles in the presence of gravitational fields 288
110. Momentum and mass of a particle. Gravitational force . . 290
111. Total energy of a particle in a stationary gravitational field . 294
112. General point mechanics ..... 295
1 13. Time-orthogonal systems of coordinates. Elimination of the dyna-
mical potentials . . . . . .296
114. Mechanics of continuous systems . . 298
115. The electromagnetic field equations . . . 302
116. Electromagnetic force and energy-momentum tensor . . 305
117. Propagation of light in a static gravitational field. Format's
principle ....... 308
CHAPTER XI. THE FUNDAMENTAL LAWS OF GRAVITATION IN
THE GENERAL THEORY OF RELATIVITY . . .310
118. The gravitational field equations .... 310
119. The linear approximation for weak fields . . .313
xii CONTENTS
120. Simple applications of the linear equations for weak fields. The
relativity of centrifugal forces and Coriohs forces . .315
121. Equivalent systems of coordinates. Systems with spherical sym-
metry . . . . . . .321
122. Static systems with spherical symmetry . . . 323
123. Schwarzsch ild's exterior solution .... 325
124. Schwarzschild's solution for the interior of a perfect fluid . 328
125. The variational principle for gravitational fields . . . 333
126. The laws of conservation of energy and momentum . . 337
127. Different expressions for the densities of energy and momentum 341
128. The gravitational mass and total energy and momentum of an
isolated system ...... 342
CHAPTER XII. EXPERIMENTAL VERIFICATION OF THE GENERAL
THEORY OF RELATIVITY. COSMOLOGICAL PROBLEMS . 346
* 129. The gravitational shift of spectral lines . . . 346
% 130. The advance of the perihelion of Mercury . 348
131. The gravitational deflexion of light . 353
132. Cosrnological models . . . 356
133. The Einstein universe 357
134. The de Sitter universe . 362
APPENDIXES .... . 371
1. Gauss's theorem . . 371
2. The transformation equations for the four-cm rent density . . 372
3. Plane waves in a homogeneous isotropie substance . 373
4. Transformation of the gravitational field variables y lic , y t , ^, a> lK by a
change of coordinates inside a definite system of reference . 374
5. Dual tensors in a three -dimensional space . . 375
6. The condition for flat space .... . 376
7. The action principle and the Hamiltoman equations for a particle
in an arbitrary gravitational field ..... 378
8. The connexion between the determinants of the space-time metric
tensor and the spatial metric tensor . . . .381
9. The derivatives of the function with respect to g%* and c? m and some
identities containing these derivatives . . .382
AUTHOR INDEX ...... 384
SUBJECT INDEX .... 385
THE FOUNDATIONS OF THE SPECIAL THEORY
OF RELATIVITY. HISTORICAL SURVEY
1. The relativity principle of mechanics. The Galilean trans-
formation
THE special theory of relativity which was developed in the beginning of
the twentieth century, especially through Einstein's work, has its roots
far back in the past. In a way, this theory can be regarded as a continua-
tion and completion of the ideas which have been the basis of our descrip-
tion of nature since the times of Galileo and Newton. The basic postulate
of this theory, the so-called special principle of relativity,! had already
in Galileo's and Huyghens's works played a decisive role in the develop-
ment of the fundamental laws of mechanics. Also the validity of the
principle of relativity for the phenomena of mechanics is a simple
consequence of the Newtonian laws of mechanics. Since the laws of
mechanics are especially well suited for the illustration of the principle
of relativity, we shall start by considering purely mechanical phenomena.
According to Newton's first law, the law of inertia, a material particle
when left to itself will continue to move in a straight line with constant
velocity. Since one cannot simply speak of motion, but only of motion
relative to something else, this statement has a precise meaning only
when a certain well-defined system of reference has been established
relative to which the velocity of the particle is assumed to be measured.
Therefore Newton introduced the notion of the 'absolute space', repre-
senting that system of reference relative to which every motion should
be measured. Experience shows that the fixed stars as a whole may
be regarded as approximately at rest relative to the 'absolute space',
for a body sufficiently far away from celestial matter always moves with
uniform velocity relative to the fixed stars.
It is, however, obvious that the law of inertia holds also in every other
rigid system of reference moving with uniform velocity relative to the
'absolute' system, for a free particle will also be in uniform translatory
motion with respect to such a system. All systems of reference for which
the law of inertia is valid are called systems of inertia. They form a
t When reference is made to the principle of relativity m Chapters I-VII we always
have in view the principle of special relativity as contrasted with the principle of general
relativity which is the basis of the general theory of relativity.
3595.60 r>
2 FOUNDATIONS OK SPECIAL THEORY OF RELATIVITY I, 1
threefold infinity of rigid systems of reference mo\ing in straight lines
and with constant velocity relative to each other. One of them is the
absolute system which is at rest relative to the fixed stars as a whole;
but as regards the validity of the law of inertia, all systems of inertia are
completely equivalent.
Now, the principle of relativity in mechanics states that the systems
of inertia are also completely equivalent with regard to the other laws of
mechanics. If this is true all mechanical phenomena will take the same
course of development in any system of inertia so that it is impossible
from observations of such phenomena to detect a uniform motion of the
system as a whole relative to the 'absolute' system. Thus, a study of
mechanical phenomena alone can never lead to a determination of the
'absolute' system.
We shall now see that the fundamental equations of Newtonian
mechanics actually are in accordance with the principle of relativity.
Let us consider two arbitrary systems of inertia, / and 1' ' . In each of
these frames of reference we use definite systems of coordinates 8 and #'.
We may, for instance, choose Cartesian coordinates x = (x,y,z) and
x' (jc',y',z'} in / and 1', respectively. According to the conceptions
of space and time, derived from our usual experience, which also form
the basis of the Newtonian formulation of the fundamental laws of
mechanics, the connexion between the coordinate vectors x and x' for
one and the same space point in the two coordinate systems JS and $'
by ' ; _ o
where v is a vector denoting velocity and direction of motion of ft'
relative to *V. / is the time and, for the sake of simplicity, it is assumed
that the origins of the two systems of coordinates coincide at the time
t -- 0. To the equations (1) may be added the equation
t' - t (16)
which states that the parameter describing the time is the same in all
systems of inertia. Thus, in the Newtonian description of physical
phenomena the time is an absolute quantity. The equations (1(7) and
(l/>) are often referred to as the Galilean transformation.
If the directions of the axes of the two systems of coordinates are
parallel, and if v has the direction of the x-axis, we obtain a special
Galilean transformation which can be written
I, 1 HISTORICAL SURVEY 3
Since the systems of coordinates 8 and 8' are completely equivalent, at
any rate as far as kinematics is concerned, and*since S obviously moves
with the velocity v relative to S f , the inverse transformations to (1)
and (2) are simply obtained by interchanging the primed and the
unprimed variables and simultaneously replacing v by v.
Let us now consider an arbitrary motion of a material particle. By
differentiation of (1 a) we get
dx' _ Jx
"eft' ~~ dt~ V
or u' = u v, (3)
where u and u' represent the velocities of the particle in the two systems
of inertia. (3) is the usual addition theorem of velocities. For a special
Galilean transformation (2), (3) reduces to
u x = u x v, u y = u y , u z = u z . (4)
When the velocity vector u, arid thus also u', is perpendicular to the
2-axis, (4) may be written
u'
u' sm&' = u$in&,
where $ and &' are the angles between the a;-axis and the directions of u
and u', respectively. Further, u ~ |u |, u' -- |u' \ denote the magnitudes
of the vectors u and u'. If we now divide one of these equations by the
other we obtain
tan,?' = -- "--V, (5)
COS IT 1 V/U
and by summation of the squares of the equations we get
( r ?; 2 H -Y
u' = u ]_2- l co8* + ^ s . (6)
( U U")
Now let us assume that the material particle with the mass m is acted
on by a force F. In the absolute system of coordinates S the particle
will then obtain an acceleration given according to Newton's second law
by the equation
From (1 a) and (16) it now follows that
&* - ^
dt'* dt 2 '
4 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 1
and .since in Newtonian mechanics forces and masses are absolute
quantities, i.e. p , = p m/ = ^ (g)
r/ 2 x'
we obtain w' ------ = F'. (10)
a/ ^
Thus we see that the second law of Newton is valid in every system
of inertia in accoi dance with the principle of relativity. This can be
expressed more accurately by stating that the Newtonian fundamental
equations are invariant under (Jalilean transformations. As is well
known, this in variance does not hold for nioie general transformations
leading to accelerated systems ot reference. If one wants to treat
mechanical phenomena in such systems, one has to introduce extra
fictitious forces, e g. centrifugal forces and Coriolis forces which only
depend on the acceleration of the frame of reference and therefore are in
no causal relationship with the physical properties of other terrestrial
systems. It was just this difference between the uniformly moving and
the accelerated systems of reference which led Newton to the conception
of absolute space.
2. The special principle of relativity
As already mentioned, the validity of the principle of relativity in
mechanics prevents a unique determination ot the absolute system of
reference from studies of mechanical phenomena alone. Now the basic
assumption of the special theory of relativity is that the special principle
of relativity is valid for all physical laws."\ According to this theory, all
physical phenomena should have the same course of development in all
systems of inertia, and observers installed in different systems of inertia
should thus as a result of their experiments arrive at the establishment of
the same laws of nature.
It this is so, the notion of absolute space obviously loses its meaning,
since any system of inertia with equally good reason can claim to be the
absolute system of reference. Of course nobody can prevent us from
calling one definite system of inertia, e g. the one which is at rest relative
to the fixed stars, the absolute system and expressing all laws of nature
in coordinates of this system. Such a procedure is, however, extremely
unsatisfactory in view of the arbitrariness in the choice of the absolute
system. It is, furthermore, very inconvenient to proceed in this manner.
The physical experiments from which the laws of nature are derived are
usually not performed in a system of reference which is at rest relative
f With tho exception of the la\vn of gravitation which find then natural place in the
general theory of lelativity.
I, 2 HISTORICAL SURVEY 5
to the fixed stars. On account of its motion around the sun the earth will
in the course of a year represent widely different systems of inertia if we
disregard the small acceleration of the earth in this motion. The trans-
formation to the coordinates of the 'absolute' system is therefore rather
complicated.
The validity of the principle of relativity for all physical phenomena
now makes such a transformation unnecessary, since the system of
inertia in which the earth is at rest at the moment considered is equiya-
lent to any other system of inertia. This obviously leads to an enormous
simplification in our description of nature.
However, this simplification has to be paid for, as we shall now see, by
an abandonment of our usual notions of time and space. The extension
of the principle of relativity to electromagnetic phenomena means, as
mentioned before, that physicists who have established their laboratories
in two different systems of inertia will, as a result of their experiments,
be led independently to Maxwell's fundamental equations of electro-
dynamics. These equations contain a universal constant c which can
be determined by means of purely electromagnetic measurements, and
is very closely equal to 3xl0 10 cm. /sec. f On the other hand, it is a
simple consequence of Maxwell's equations that electromagnetic waves
in empty space propagate with the velocity c, independently of the way
in which they are created. Since light waves, according to Maxwell's
theory of light, are special electromagnetic waves, the velocity with which
light is propagated in vacua must also be independent of the state of
motion of the light source and equal to the constant c. If Maxwell's equa-
tions in accordance with the relativity principle are valid in any system
of inertia, the velocity of light must have the same constant value c in
all systems of inertia, independently of the motion of the light source.
This is obviously in conflict with the usual kinematical concepts accord-
ing to which we should expect, for instance, to find a lower velocity of
light in X' than m f} if the relative motion of >V with respect to H has the
same direction as the direction of propagation of the light ray.
Consequently the acceptance of the relativity principle must neces-
sarily lead to a revision of our ordinary concepts of space and time.
Before taking such a radical step one would naturally w r ant to be sure
that it is really necessary. This question can only be settled as a result of
experiment. Optical experiments are especially suited to this purpose
in view of the high accuracy obtainable with optical instruments. In
t W. Wober und R. Kohlrausch (1856), Ostwalds Klassiker der exakten Wis&en-
schaften, No. 142.
6 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 2
the following section we shall therefore give a short historical survey of
the numerous optical experiments which have been performed in an
attempt to detect effects depending on the motion of the apparatus with
respect to an 'absolute' space. These experiments all gave negative
results and finally led to a general acceptance of the principle of relativity .
3. Invariance of the phase of a plane wave
In contradistinction to the relativistic standpoint according to which
Maxwell's equations are valid in any inertial system, Maxwell and his
contemporaries maintained that the fundamental equations of electro-
dynamics were valid only in one system of inertia, \ iz the system which
is at rest relative to the so-called 'world ether'. The ether was imagined
as a medium which penetrates through all matter and empty space and
which was the carrier of all optical and electromagnetic phenomena
Moreover, the ether was supposed to lepresent the absolute system of
reference, thus giving a substantial physical meaning to Newton's notion
of absolute space. In the present section we shall fully adopt this point
of view, and our first task will be to see what consequences this will have
for the course of development of optical phenomena in a system of inertia
moving relative to the ether.
Let 8 be a Cartesian coordinate system which is at rest in the ether.
Relative to Ft, a plane monochromatic light wave in empty space will
have the propagation velocity c 3 X 1 10 cm /sec. A wave of this type
is completely determined by the phase velocity, the frequency of the
wave, and the direction of propagation. In the first place, we shall find
the transformation of these three quantities by a transition to a co-
ordinate system *S' moving relative to the ether with a constant velocity
v in the direction of the .r-axis.
For simplicity let us assume that the normal of the vvavo plane lies in
the .r//-plane. Then the* wave is described in N by a wave function
ifj .4 cos27r/< T ,
A (11)
where v is the frequency and ex is the angle between the wave normal n
and the .s-axis. / .rcosa+j/sincx is the distance from the origin
to that wave plane which contains a point p with the coordinates (r, y)
(cf. Fig/1).
The phase F in (11) has the following simple physical meaning.
Let us assume that the wave crest which passes the origin at the time
I, 3
HISTORICAL SURVEY
t = is provided with a label. Now suppose an observer to be placed at
the point p who at the moment when the labelled wave arrives at p
begins to count the waves passing over the point p. The number of waves
counted by the observer up to the time t will then just be equal to the phase F.
In fact, v waves arrive per second and, since the labelled wave takes l/c
seconds to move from O to p, the observer is counting during an interval
of / (1/c) seconds.
Fir;. 1.
Now let 8' be the moving system introduced above, and let us assume
that the two coordinate systems 8' and 8 coincide at the time t -
when the labelled \vave passes the common origin of the systems. If;/
is a point in 8' with coordinates (x\ ?/), which coincides with;? at the time
t t', the number of waves passing // trom the time of arrival at;/ of the
labeljed wave up to the lime t will of course be the same number F as
before. On the other hand, this number counted by an observer at p'
will by a similar argument to that applied in 8 be equal to
F ^ v'\t' ~
V
,i , x cos a )-?/ sin a
c'
where the primed letters in (12) denote the same physical quantities as
the corresponding unprimed letters in (11), but now measured in the
system of coordinates S' . Thtfb the phase F is an invariant.
8 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 4
4. Transformation of the characteristics of a plane wave
The connexion between the coordinates (.r, ?/, /) in (11) and the co-
ordinates (#', ?/, t f ) in (12) is obviously given by the Galilean transforma-
tion (2), since the pointy/ coincides with p at the time t' t. Equating
the two expressions (11) and (12) for F and eliminating the coordinates
(x,y,t) by means of (1) we obtain
a;' cos '+?/' sin nA
Now this equation must hold for all values of the independent variables
x', y', t', and this is possible only when the coefficients of these variables
are equal on both sides of the equation (13). Consequently we get the
equations / v ,
v = vll cosaj, (14)
\ c I
v Sin a
(is)
v COS a v COS c\
From equations (15) we get at once
tan QL = tan a, (16)
i.e. a' a, (17)
and, further, ~ - ~. (18)
c " c
By solving the last equation for c! we find by means of (14) that
c' c~ vcosoc. (19)
The equations (14), (17), and (19) show how the three characteristics
of the wave, viz. the frequency, the direction of the wave normal, and
the phase velocity, will change at the transition to a coordinate system
in uniform motion with respect to the ether. Equation (17) shows that
the direction of the wave normal is the same in both systems of inertia.
On the other hand, if a -/ \TT, the equations (14) and (19) involve the
velocity v, so that a measurement of frequency and velocity in principle
should be suited to determine the motion of the laboratory system with
respect to the ether. In the following we shall discuss these two effects
separately.
5. The Doppler effect
Equation (14), which is the mathematical expression of the so-
called Doppler effect for light waves, gives the connexion between the
I,5 HISTORICAL SURVEY 9
frequency v' in a moving frame of reference and the 'absolute' frequency
v as observed by an observer at rest in the ether. If n denotes a unit
vector in the direction of the wave normal, and v is the velocity vector
of S' relative to the ether, (14) can also be written
(20)
where n. v is the scalar product of the two vectors. The Doppler effect
appears when the observer is moving relative to the source of light.
However, the formula (20) cannot be used directly in such a case, since
as a rule both the observer and the source of light will have a motion
relative to the ether. If v is the proper frequency of the light source,
i.e. the frequency measured by an observer at rest relative to the source
of light, we have in analogy to (20)
v-Kl--n.v/c), (21)
where v is the velocity of the source relative to the ether.
By elimination of the unknown absolute frequency v we obtain from
l-(n.v)/c
The frequencies v and i>, the direction of propagation of the light n,
and the relative velocity v r v v of the observer with respect to the
source of light can be determined directly by experiments, and (22) then
permits in principle a determination of the absolute velocities v and v
of the light source and the observer.
Both v and v are, however, very small compared with r, so that we
may perform an expansion of (22) in terms of the small quantity (vn)/c.
If we neglect all terms oi higher than second order in v/c and v/r, we get,
introducing the relative velocity v r v v,
(n^)_(n^^
c c 2 j
In iirst approximation, the Doppler effect thus depends only on the
relative velocity v r . The absolute velocity v of the light source enters
only in the small second-order terms.
The Uoppler effect is observed in the spectra of the stars, the lines of
the spectrum being shifted towards violet or red according as the earth,
during its annual motion, moves nearer to or away from the observed star.
The velocity of the earth in its orbit is approximately 3x 10 6 cm./sec.
10 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 5
and cosmic velocities are mostly of the same order of magnitude. Con-
sequently, we have v/c 10~ 4 , i.e. the terms of the second order will be
of the order of magnitude 10~ 8 , which is far beyond the precision of such
measurements.
The Doppler effect has been observed also in the light from moving
terrestrial sources. By measuring the frequency of the light emitted by
rapidly moving hydrogen molecule ions in a positive ray tube, J. Stark f
found good agreement with the formula (23) as regards the terms of the
first order. In these experiments the relative velocity v r and, thus, also
v were of the order of 10 8 cm. /sec., i e. r/c & gJ . Also in this case, the
second-order terms were too small to be measured, and these experi-
ments did not therefore allow a determination of the absolute velocity.
Much later, in the thirties, such experiments were repeated by I vest
with an improved experimental technique, which allowed the second-
order terms to be determined. The results obtained were not in agreement
with (23), but they agreed with a formula derived from the theory of
relativity (equation (II. 90), ('hap. II, 25). The second-order term was
found to be independent of the direction of the light emitted and depen-
dent only on the relative velocity v r . No motion relative to the ether could
be observed, in agreement with the principle of relativity.
These experiments, which were performed much later, had of course
no influence on the historical development of the relativity theory.
Because of the limited accuracy, the experiments by Stark did not allow
any decision to be formed on the validity of the principle of relativity,
but on the other hand the results of these experiments were not in contra-
diction with the principle of relativity.
6. The velocity of light in vacua
We now turn to the question whether a measurement of the velocity
of light by terrestrial methods can lead to a determination of the absolute
velocity v of the earth. Since v enters into equation (19), which can also
be written , . x /0/1X
c' = c (n.v), (24)
this should be possible in principle, as mentioned on p. 8. The well-
known measurements of the velocity of light by Fizeau (1849) and Fou-
cault (1865) showed, however, no influence at all of the motion of the
earth. The velocity of light was always found to be the same, in agreement
f J. Stark, Ann. d Phys 21, 40 (1906), J. Stark and K Siogel, ibid 21, 457 (1906),
J Stark, W. Hermann, and S. Kmoshita, ibid 21, 462 (1906).
J Cf , for example, H. E Ives and G. R Stilwell, Journal of the Optical Society of
America, 28, 215 (1938).
I, 6 HISTORICAL SURVEY 11
with the principle of relativity. How can this result be understood on
the basis of the ether theory ?
In the first place, it should be observed that what is measured in these
experiments is not the phase velocity but the so-called ray velocity. In
Fizeau's original experiments, for example, a light signal is sent along a
certain path and back again, and the difference between the time of
departure and the time of return of the signal is measured. The velocity
is then determined as the ratio between the length of the path traversed
and this time interval. Now it is true that the velocity of the light signal
is equal to the phase velocity c in the coordinate system 8 which is at
rest in the ether, but in the moving system 8' the velocity of the signal
will not be equal to the phase velocity r/ given by (24). This is plausible
if one keeps in mind that a light signal represents a certain amount of
electromagnetic energy, and that energy, like mass, is a quantity which
is conserved, so that a signal in many respects will resemble a material
particle. Therefore we should rather expect that the velocity of a light
signal in a moving system of coordinates S' is given by the equations
(3), (5), and (6) if in these equations u is put equal to c, the velocity of
light in the ether.
A closer treatment based on the wave theory of light confirms this ex-
pectation. Such a treatment shows that in a moving coordinate system
the ether acts like an anisotropic medium so that one has to distinguish
between the phase velocity and the ray velocity, which is identical with
the velocity of propagation of the light energy and is just given by (3).
To see this we shall apply the well-known Huyghens principle which for
all phenomena in the domain of geometrical optics is a consequence of
Maxwell's equations. In accordance with this principle we obtain the
consecutive wave surfaces as the envelopes of elementary waves starting
from each point of a wave surface.
Let us consider the propagation of light in a system of coordinates 8'
moving with the velocity v relative to the ether system S. In S' we thus
have an 'ether wind' of the velocity v which will carry along the ele-
mentary waves in the same way as sound waves are carried by the
wind.
Fig. 2 gives a diagram of successive positions of light waves in the
system 8'. Let the surface a denote the position of a given wave front
at the time t. In order to construct the wave front a 1 at the time t-\-dt
we regard every point P on a as a starting-point of an elementary wave.
Because of the ether wind, this elementary wave will at the time / \-dt
obviously form a sphere E with centre at a point Q which lies at a distance
12 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 6
v dt from the point P in the direction of the ether wind. The infinitesimal
vector PQ is thus given by
PQ = -vdt. (25)
Since the velocity of propagation of the elementary wave in the ether
is c, the sphere E has a radius QP^ = c dt. The wave front o is now
FIG 2.
obtained as the envelope of all the elementary waves, i.e. the vector QP l
is perpendicular to a 1 at the point of contact P v and in the limit as dt ->
QP l is also perpendicular to or. Thus, the vector QP l lies in the direction
of the phase-velocity vector, and we have consequently
^.
\ = c dt = en dt,
(26)
where c = nc is the phase-velocity vector in the ether. In S' the phase
velocity is by definition given by
PA = c'dt = c'n' dt,
(27)
where the unit vector n' denotes the direction of the wave normal in S' .
Obviously n , = n (2g)
in accordance with (17). Since any infinitesimal part of a curved wave
surface can. be regarded as plane, the connexion between c' and c must
again be given by (24). This follows also directly from (27), (28), (26),
and (25) if we note that
PB = QP l = c dt,
while BA is equal to the projection of the vector BP l PQ v dt
on the direction n.
I, 6 HISTORICAL SURVEY 13
The relative direction of the ray, i.e. the direction of propagation of the
light energy as estimated by an observer in $', is now given by the direc-
>
tion of the vector PP V and if u' denotes the relative velocity of the ray
we have >
PP l = u' dt = u'e' dt, (29)
where e' is a unit vector indicating the relative direction of the ray.
U
V
FIG. 3
Now the vector PP l is the sum of the vectors PQ and Ql\
> > - ->
In the limit as dt -> this gives, because of (25), (26), and (29),
u' = c v. (31)
In the absolute system the ray velocity is identical with the phase velocity
u ue c en, (32)
i.e. u = c, e = n = n',
so that (31) can be written
u' = u v. (33)
Thus we obtain the same addition theorem for ray velocities as for
particle velocities- u is the geometrical sum of u' and v. If & and &'
are the angles between the direction of the velocity vector v and the
absolute and relative ray directions, respectively, a consideration of
the triangle in Fig. 3 gives at once, since u - f ,
, (34)
-v/c' { )
and u' 2 +v 2 + 2vu' cos #' = u 2 - c 2 .
By solving this equation with respect to u' we get
= {c 2 -t; 2 +(v.e') 2 }*-(v.e'). (35)
14 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 6
A comparison between (35) and (24) shows that the relative ray velocity
in general is different from the relative phase velocity, the difference
being of the second order in v/c. Only when the direction of the ray
(and the direction of the wave normal) is equal or opposite to the direc-
tion of v, are the two velocities identical and equal to cv and c+v,
respectively.
Now it is clear that the velocity measured by Fizeau's and Foucault's
methods is the ray velocity, but, since v also enters into (35), it should be
possible in principle to determine the absolute velocity of the earth on
the basis of these measurements. It is, however, easy to understand
why no variation in the velocity of light was ever observed. In these
experiments a light ray is sent along a known closed path and the time
which the light signal takes to travel along this path is measured. In
order to make the path as long as possible within a limited space, the
light ray is reflected many times by suitably arranged mirrors. Let
^i> ^25-"> ^ be the distances traversed by the ray between the mirrors, and
let the corresponding directions of the ray be given by the unit vectors
e i> e/ 2v> e l; then we obviously have
since the ray describes a closed polygon. The time needed for the light
to traverse this closed path is then, according to (35),
t = ^C l i / 37 \
^{ C 2_ v 2 +(v . e ; )2 p_ (v . e ;j'
If this expression is expanded in terms of the small quantity v/c we get,
neglecting terms of order higher than the first,
t =
On account of (36), the first-order term disappears and in this approxi-
mation we obtain
Thus, when terms of order higher than the first are neglected, the
measured time is the same as if the earth were at rest in the ether. A
determination of the absolute velocity v would consequently require a
measurement of quantities of at least second order. Fizeau's and
Foucault's methods, however, did not allow such a high accuracy, and it
is therefore understandable even on the basis of the ether theory that
the experimental results * were in agreement with the principle of
I, 6 HISTORICAL SURVEY 15
relativity. It was not until many years later that Michelson was able
to develop a method which also allowed the measurement of magnitudes
of the second order and thereby gave a final proof of the validity of the
principle of relativity. We shall, however, follow the historical trend of
development and return to a discussion of Michelson 's experiments in a
later section ( 12).
7. The velocity of light in refractive media
Hitherto we have only discussed the propagation of light in empty
space. Let us now assume that the space is filled with an isotropic
transparent substance with the index of refraction n. If the substance
is at rest relative to the world ether, the phase velocity in the absolute
system S is, according to Maxwell's phenomenological electrodynamics,
r 1== |, n = M ! , (38)
where e is the dielectric constant of the medium and /t its magnetic
permeability The phase velocity c\ relative to a moving system of co-
ordinates is then, in analogy to (-4), given by
^-^-(n.v). (39)
This formula is valid if the refractive body is at rest in the absolute
system 8 But suppose the body moves \\ith a velocity v, thus being at
rest in 8' , what will then be the expression for the phase \ elocity in S' ?
This problem was much discussed in the early days. The simplest
assumption is that (39) remains valid, i.e that the ether passes un-
disturbed through the moving body without being dragged along. Then
we have in ft' an ether wind with the velocity v, and we can now find
the ray velocity by means of Huyghens's principle in the same way as in
6. In equation (35) wo have simply to replace c by the phase velocity
c t ^ c/7i and thus get the following expression lor the ray velocity in S' :
u' -- [c\ -v*+(v.e')*}i- (v.e'). (40)
Instead of assuming that the ether passes undisturbed through the
moving body, it has also been suggested that the ether is completely
dragged along by the body. According to this hypothesis, which was put
forward by Stokes, | we obviously get
u' = c[ = 2 9 (41)
for, in this case, there would be no ether wind in S'.
t G. G Stokos, Phil Mag (3), 27, 9 (1845), Mathematical and Physical Papers, 1,
134 (1880).
16 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 7
A third possibility is to assume that the ether is dragged along only
partly by the moving body, say with a velocity av, where the 'dragging
coefficient' a is a positive number smaller than 1 depending on the re-
fractive index n. This hypothesis was put forward by Fresnelf who,
on the basis of the elastic ether theory, gave the following expression for
the dragging coefficient : ,
=!--,. (42)
On this hypothesis the velocity of S' relative to the dragged ether is
v v = (v/w 2 ) and, instead of (39), we obtain for the relative phase
velocity , }
4 = 4-^. ()
where c* = c/n is the phase velocity in a system of coordinates S*
accompanying the dragged ether. Consequently the phase velocity in
the absolute system S is
Ci - c*+ a (v.n) = f + ( v .n)/l-l), (44)
since S is moving with the velocity cxv relative to S*.
In order to find the relative ray velocity u' by means of the method
outlined in 6, we must let the system of coordinates 8* take over the
role played by S in the former considerations. Since the ether wind in S'
has the velocity (v/n 2 ), we get, in analogy to (31),
u' = C?-^, (45)
n*
where the vector c* with the magnitude c* = c/n is the phase-velocity
vector in S*. The magnitude of the relative ray velocity we obtain in
the same way from (35) by replacing c and v by c* c/n and (v/n 2 ),
respectively, i.e.
In the absolute system S, however, we have an ether wind with the
velocity av = (1 l/n 2 )v. Therefore we have for the absolute ray
velocity u, in analogy to (45) and (46),
u = cf+av, u - {cf 2 -aV+a 2 (v.e) 2 }+a(v.e))
1 1 , c <)
p^ . . J * ... ^ 1
n 2 ' l n' '
| A. J. Fresnel, Ann. de chim. et de phys. 9, 57 (1818).
I, 7 HISTORICAL SURVEY 17
or, neglecting terms of the second order in v,
w = - + a(v.e). (48)
n
By elimination of cj from (45) and (47) we obtain again the simple
addition theorem
u == u >_|_ y
A comparison between the equations (43), (46), (44), and (47) shows
that the ray velocities are identical with the phase velocities if the direc-
tion of propagation of the light is the same as or opposite to the direction
of the velocity v. Without any calculation this follows immediately from
the fact that, in this case, the ether wind carries the elementary waves
in a direction parallel to the light beam. The considerations given in
this section are valid also for the case of in homogeneous bodies with a
continuously varying index of refraction. Only in this case the system
$*, which depends on the value of n, will be different at different points
of the substance.
8. Hoek's and Fizeau's experiments
A measurement of the velocity of light in transparent substances seems
to offer a new possibility for a determination of the absolute motion of
the earth. An experiment of this kind was performed in 1868 by Hoekf
who used an interferometer arrangement of the type shown in Fig. 4.
A monochromatic light ray from a source of light L is divided by a
(weakly silver-coated) glass plate P which is placed at an angle of 45
into a transmitted part 1 and a reflected part 2. The transmitted ray 1
is reflected by the mirrors S l9 $ 2 , $ 3 and traverses a rectangular path
P/S^/Sg/S^P; again a certain fraction of the ray passes the plate P and
enters the telescope T. The reflected ray 2 traverses the same rectangle
in the opposite direction. On its return to P it is partly reflected into T
where it interferes with 1 . Between S 2 and $ 3 is inserted a tube of length I
filled with a substance with refractive index n (for instance water).
Even if the whole apparatus were at rest in the ether, such an arrange-
ment would give rise to interference fringes in the telescope, since the
slope of the mirrors cannot possibly be adjusted so accurately that two
rays 1 and 2 which focus on the same point in the telescope have traversed
a path of exactly the same optical length. However, if the whole appar-
atus has a velocity v with respect to the ether, this will cause an extra
phase difference &F between the rays 1 and 2, which can be calculated
by means of (40), (41), or (46).
t M. Hook, Archives Nderlandaises des Sciences Exactes et Naturelles, 3, 180 (1868),
3595.60 n
18 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 8
Let us, for simplicity, assume that the apparatus is set up in such a
way that the lines PS l and$ 2 S 3 are parallel to the direction of motion v
of the apparatus relative to the world ether. The phase difference A F
resulting from the absolute motion of the apparatus will then obviously
be due to a difference in the times t l and t 2 which the rays 1 and 2, respec-
tively, require to traverse the tube of length A B and the corresponding
distance CD on the path connecting P and /S\. On the remaining paths
DS 1 S 2 B and A$ 3 PC the two rays are completely equivalent so that no
contribution to the phase difference &F can arise from these parts.
. - L - >
i
i v
A
r i
B X
L /
P J
C
n
> /^ t
FIG 4.
The values of the quantities t and 2 will now depend on the degree
to which the ether is dragged along by the refractive substance. If there
is no dragging at all we get, according to (40),
1 cv c/n-{-v' 2 c/nv c+v'
where we have put the refractive index of air equal to 1. The phase
difference between the rays 2 and 1 , in so far as it is due to the absolute
motion of the apparatus, is then
AJF = v^-tj = 2lw
(c/n) 2
cv
or, if we neglect terms of order higher than the first in v/c,
C C
(50)
When the apparatus is at rest with respect to the earth, v is identical
with the absolute motion of the earth. If, therefore, the apparatus were
rotated 180 around an axis perpendicular to the direction of motion of
the earth, the phase difference in question would be AjF. Such a
rotation should thus cause a shift of the interference lines corresponding
I, 8 HISTORICAL SURVEY 19
to a phase shift of 2&F and, since Af according to (50) is a quantity of
the first order in v/c, this effect should be easily observable.
The result of Hoek's experiments was, however, negative; no observ-
able shift of the interference lines could be detected after a rotation of the
apparatus. This result too is in complete agreement with the principle of
relativity according to- which all phenomena should be independent of
the state of motion of the measuring instruments.
Since the accuracy of Hoek's experiments did not go beyond terms of
the first order, the negative result of this experiment was, however, not
a serious difficulty for the ether theory; it only showed that the equation
(40), based on the assumption that the ether is not dragged along by the
refractive body, could not be maintained. Similarly, Stokes 's hypothesis
is ruled out by the result of Hoek's experiment, for from (41) we would
getf i;A/f1 AF=-^ (51)
C C
On the other hand, the formulae (43) and (45), corresponding to a
dragging coefficient (42), are seen to give an explanation of the result
obtained by Hock. In this case, we get, by means of (45) or (46),
AL1 / I I I I \
]i _ yj I I
\c/n v/n 2 c-\-v c/n-\-v/n 2 c v]
(52)
and if we neglect terms of order higher than the first, this quantity is zero.
It is also seen immediately that only Fresnel 's value (42) for the dragging
coefficient oc gives a zero value for AF in first approximation. Hoek's
experiment can therefore be regarded as an experimental verification of
Fresnel's formulae for the velocity of light in moving bodies, at least as
regards terms of first order.
As early as 1851, Fizeau j had obtained the same result by measuring
the velocity of light in running water. The experimental arrangement
was very similar to the interferometer arrangement in Hoek's experi-
ment (see Fig. 5). The only difference is that the light rays 1 and 2 here
are passing through water on the path PS l as well as on the path S 2 S^.
As indicated in the figure, the ray 1 is traversing the water in a direction
opposite to the direction of motion of the water, while the ray 2 has the
same direction of motion as the water. Now Fizeau compared the posi-
tion of the interference fringes while the water was at rest in the tubes
t H Fizeau, C.R. 33, 349 (1851) ; A. A. Michelson and E. W. Morley, Amer. Journ. of
Science, 31, 377 (1886); H. Fizeau, Ann. d. Phys. und Chem., Erg. 3, 457 (1853).
20 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 8
with that when a strong water current was sent through the tubes. A
marked shift of the fringes could be observed.
From Hoek's experiment we know that the motion of the earth rela-
tive to the ether can have an effect of second order only; we may therefore
in our calculation make the assumption that the apparatus is at rest in
the absolute system 8. The velocity vector v in (48) is then simply equal
\s
T
A
C
B
D
to the velocity of the water relative to the tube. Since e is parallel to v
on the paths A B and CD, which are the only paths which give rise to a
phase difference between 1 and 2, we obtain from (48) for this phase
difference
where I means the whole path through which the light ray travels in the
water. The shift in the position of the fringes observed by Fizeau was in
complete agreement with the phase difference given by (53).
9. Lorentz's theory of electrons
The experiments mentioned in the preceding section may be regarded
as a decisive experimental verification of Fresnel's formulae (42)-(48),
at least as regards all terms of first order. However, the derivation of
these formulae from the point of view of the primitive ether theory
meets with a serious difficulty when we keep in mind that the index of
refraction generally depends on the frequency of the observed light.
Since the dragging coefficient a in (42) is a function of n, this would mean
that the dragging of the ether is not only dependent on the properties
of the moving body, but also on the frequency of the light. Strictly
I, 9 HISTORICAL SURVEY 21
speaking, one would have to introduce a separate ether for each colour
of the light.
This is, of course, an impossible assumption, and this difficulty imder-
lies both Fresnel's mechanical ether theory and Maxwell's phenomeno-
logical theory in which the ether represents the system of reference where
Maxwell's equations are valid. In fact, the dependence of the dragging
phenomena on the frequency makes it impossible to decide in which
system of reference the equations of Maxwell's electrodynamics are valid.
This difficulty is closely connected with the fact that the index of
refraction in this theory is constant and equal to (eju)* and thus does not
give any explanation of the dispersion phenomena. A satisfactory
explanation of the dispersion and of the dragging phenomena was given
by Lorentz f in his theory of electrons. According to this theory, the
ether is not dragged at all by refractive substances, but stays constantly
at rest in a certain system of inertia the absolute system. The material
bodies are assumed to be composed of atoms which contain a number of
positively and negatively charged electric particles. While the positive
particles contain practically the whole mass of the atom, the negative
particles, the electrons, are supposed to be very light. Under the influence
of the electromagnetic fields in a light wave they perform forced vibra-
tions around their equilibrium positions. Therefore the electrons them-
selves will emit electromagnetic waves which interfere with the incident
wave in such a way that the effective velocity of propagation of the light
in a medium at rest is c/n instead of c.
According to this theory it is also clear that the coefficient n in general
depends on the position of the frequency of the incident wave relative to
the proper frequencies of the electrons. Furthermore, Lorentz was able
to show that a uniform motion of the refractive body modifies the waves
emitted by the vibrating electrons in such a way that the effective phase
velocity of the light in the moving body to a first approximation is given
by Fresnel's formulae (43) and (44).
Thus as regards the propagation of light in refractive bodies Lorentz 's
electron theory gave, at least to a first approximation, the same results as
Fresnel's theory, avoiding, however, the serious objections which could
be raised against Fresnel's derivation of his formula. In one respect it
even gave a more precise formulation of Fresnel's formula (44). Since the
index of refraction n in dispersive media depends on the frequency v,
and since the frequencies on account of the Doppler effect are different
t H. A. Lorentz, The Theory of Electrons, Leipzig, 1916. See also L. Rosenfeld, Theory
of Electrons, Amsterdam, 1951.
22 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 9
in 8 and in 8', it must be specified which value for the frequency and thus
for n should be inserted in (44). Now Lorentz was able to show that one
must insert the value n(v'), where v is the frequency in the system 8'
moving along with the refractive body, while n is the function of the
frequency which is valid for a body at rest in the ether.
We shall not go deeper into Lorentz 's theory since all the results men-
tioned above will be derived later in a much simpler way from the theory
of relativity (Chapter II). Here we shall confine ourselves to the remark
that Fresnel's formulae (42)-(48) are a consequence of the electron theory
if we neglect all terms of order higher than the first. In the following
section we shall use these formulae to show that, for all optical effects of
the first order, the ether theory in the form given it by Lorentz yields
results which are in agreement with the postulate of relativity.
10. Agreement between the ether theory and the principle of
relativity as regards all effects of the first order. Fermat's
principle
According to the principle of relativity the track of a light ray con-
necting two points which are fixed relative to the earth should be com-
pletely independent of the absolute motion of the earth. This must at
least be true approximately, otherwise it would be impossible to make
constant optical images of objects. If the passage of the light rays
through the lens systems of optical instruments were markedly depen-
dent on the absolute motion of the earth, the image formation in such an
instrument would be time-dependent, an effect which has, however,
never been observed.
As was shown by Lorentz, f this fact can easily be explained on the
basis of Lorentz 's electron theory when we assume that all terms of
the second order are too small to be measured. This result of Lorentz 's
theory is the more remarkable as the relative ray velocity u' is, to a first
approximation, essentially dependent on the absolute velocity v. Neglect-
ing all terms of higher than the first order in (46), we get
u' .=- c/n (v.e')/n 2 . (54)
In order to construct the track of a ray on the basis of this expression
we shall again consider Fig. 2 (p. 12). As in (29), we have
t - u'e' dt,
where u' is given by (54) and e' is a unit vector in the direction of the
relative ray through the point P. But instead of (25) and (26) we have
f See ref., p. 21.
I, 10 HISTORICAL SURVEY 23
now QP l = c dt/n and PQ = v dt/n 2 , where n is the index of refraction
at the place in the medium considered, for in the present consideration
the system $*, which is at rest in the 'dragged' ether, plays the same role
as the absolute system S in vacuo. Let us call two points P and P l9
which are lying on the same ray and on consecutive wave planes,
a and a v conjugated points. Then the construction of the light ray
obviously consists in a determination of the conjugated points on the
consecutive wave planes. Now consider two arbitrary points on a and <r 1 ,
respectively, with the distance ds and let us form the quantity dsju' ',
where u' is given by (54), with the direction e' equal to the direction of the
line connecting these points. If the two points are conjugated as P and
P f in Fig. 2, ds/u' is equal to dt, where dt is the time which the wave front
takes to travel from the position a to the position a v If the two points
are not conjugated as, for instance, P and E in Fig. 2, ds/u' will always
be larger than dt. For, in this case, we have dt PR'/u' 9 where PR' is
the distance from P to the point of intersection between the line PR and
the elementary wave E, and since 11 lies outside E, we have ds >> u f dt.
Now let A and B be two fixed points in the refractive body. Consider
the integral n
'"
u
A
along an arbitrary curve connecting the points A and B, where u' at
any point of the curve is the relative ray velocity (54) corresponding to
the direction of the element ds. According to the above arguments the
integral (55) will then assume the lowest value when the curve coincides
with the light ray through the points A and B, because only in this case
will all elements ds of the curve connect conjugated points.
The ray between two arbitrary fixed points A and B in the refractive
body is thus determined by the condition that the integral (55) is a mini-
mum for the track of the ray, and since the integral is equal to the time
which the light ray needs to travel from A to J5, this theorem is identical
with Fermat's principle which is thus a consequence of Huyghens's
principle.
For the integrand in (55) we get, by means of (54), as a first approxi-
mation , , / ,v
1 = 1 = n (v.e')
u' c/n (v.e')/n 2 c c 2
B B B
Hence, J J = I J d* + Llv. j ds), (57)
A A \ A '
24 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 10
where ds e' ds is an infinitesimal vector joining two consecutive
points on the curve connecting A and B. The last term in (57) is con-
sequently equal to v/c 2 times the projection of the curve on the direction
of v and this projection is the same for all curves connecting the fixed
points A and B.
Thus, for the purpose of variation we can replace (55) by
/*-/V < 5s
A A
This expression is, however, equal to the time which the ray would need
to travel from A to B if the refracting body were at rest in the ether. To
a first approximation the light track between two points in the moving
body is thus the same as if the body were at rest, in agreement with the
principle of relativity. If the path of the ray in a medium at rest is
mapped out by means of suitably arranged screens with small openings,
the ray will also pass through these openings if the whole apparatus is
moving with constant velocity.
Hoek's experiment showed that the interference phenomenon occur-
ring in his special experimental arrangement, at least in first approxima-
tion, was independent of the absolute motion of the earth. Nor has any
influence of the absolute motion of the earth ever been detected in the
numerous later interference experiments. These facts, which are in com-
plete agreement with the postulate of relativity, can, however, as shown
by Lorentz,*)" easily be explained on the basis of the ether theory if we
may assume that terms of the second order are below the accuracy of
the experiments.
Let us consider an arbitrary interference experiment where all parts
of the apparatus are at rest relative to the system of reference S' which
follows the motion of the earth. Such an experiment always involves
two rays 1 and 2 which start from the same point A and are brought to
interference at another point B after having traversed different paths
I and II from A to B. Now, according to (57), the times t l and t 2 which
the rays 1 and 2 take to travel from A to B are
(59)
i
i(v.Jd 8
II II x II
Seeref., p. 21.
I, 10 HISTORICAL SURVEY 25
where the integrals occurring in (59) should be taken along the two paths
I and II from A to B. Since these paths have common end-points, the last
terms in the equations (59) are equal and the difference in time is simply
given by C n C n
A = 1 - 2 = -efc- (~ds. (60)
i ii
Thus the time difference is the same as if the apparatus were at rest in the
ether. Since the absolute velocity of the earth does not enter into (60) it is
obvious that the phase difference between the rays 1 and 2 at the point B,
which is obtained by multiplying the time difference by the frequency,
remains unchanged when the apparatus is rotated so as to give a different
position with respect to the direction of motion of the earth. Therefore
such a rotation (to a first approximation) will not cause any shift of the
interference fringes.
II. The aberration of light
As we have seen in the preceding paragraph, the direction of a ray of
light is, at any rate to a first approximation, independent of the absolute
motion of the light source and the observer. However, the direction of a
light ray depends essentially on the velocity of the light source relative
to the observer. This phenomenon, which is called aberration, was ob-
served in 1727 by Bradley f who noticed that the stars seem to perform a
collective annual motion in the sky. This apparent motion is simply due
to the fact that the observed direction of a light ray coming from a star
depends on the velocity of the earth relative to the star.
In order to find the magnitude of the aberration we consider a point
P' just outside the atmosphere of the earth, but in a fixed position rela-
tive to the system of reference S' following the earth. According to the
above considerations, the aberration depends only on the relative
velocity between the star and the observer, and we may therefore, for
the sake of simplicity, assume that the star is at rest in the absolute
system 8. Now we consider a light ray which comes from the star and
passes through the point P'. The absolute direction of the ray thus
determines the direction in which the star would be observed if the earth
were at rest, while the relative direction of the ray determines the
apparent position of the star. Since the index of refraction n is equal
to 1 at the point P', the connexion between the absolute and the relative
direction of the ray is given by formula (34). Let 6 and 6' be the angles
between the direction of motion of the earth and the actual and the
t J. Bradley, Phil. Trans. 35, 637 (1728).
26 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 11
apparent directions to the star, respectively. We have then & =
and #' = 77+0', and from (34) we get
tan0'= --. (61)
costf+u/c
Since now the path of the light ray from P' to the astronomer's tele-
scope on the earth, at any rate in first approximation, is independent of
the state of motion of the earth, the light will not suffer any further
aberration on this path. This is true even if the ray in the course of its
path passes through a strongly refracting medium as, for example, when
the telescope is filled with water. Such an experiment was performed by
Airy (1871),f who showed that the magnitude of the aberration was not
changed by the presence of the water. The aberration formula (61) has
proved to be in complete agreement with observations.
12. Michelson's experiment
As we have seen, the results of all experiments discussed up to now
were in agreement with the postulate of relativity , however, the accuracy
of these measurements (with the exception of Ives's experiment which
was performed much later) was not good enough to allow the measure-
ment of terms of higher than the first order. To this approximation, how-
ever, Lorentz's electron theory, which is based on the concept of an
absolute ether, was in agreement with the postulate of relativity. A
motion of the earth relative to the ether should, according to Lorentz's
theory, influence the terms of the second order only. In order to obtain
a decisive experimental test of the postulate of relativity it was, there-
fore, of the utmost importance to devise an experimental arrangement
permitting a measurement of quantities of the second order.
This was accomplished in 1881 by Michelson, J who measured the
velocity of light by means of the interferometer arrangement outlined
in Fig. 6. By means of a glass plate P a light beam from a light source L
is divided into two rays, 1 and 2, perpendicular to one another. The
transmitted ray 1 is reflected by a mirror S t back to P, where some of the
ray is reflected further into a telescope T. Analogously, the ray 2 is
reflected by a mirror S 2 back to P and part of the ray goes through the
glass plate and enters the telescope where it interferes with the ray 1 .
Even if the apparatus were at rest in the ether we should obviously
observe a set of interference fringes in the telescope.
t G. B Airy, Proc. Roy. Soc. London, A, 20, 35 (1871), 21, 121 (1873); Phil. Mag.
43, 310 (1872).
} A. A. Michelson, Amer. Journ. of Science (3), 22, 20 (1881); A. A. Michelson and
E. W. Morley, ibid. 34, 333 (1887).
I, 12
HISTORICAL SURVEY
27
Now let us assume that the apparatus is placed in such a way that the
path P/Sf x is parallel to the direction of motion of the earth in the ether,
and let the paths PS l and P^ 2 have the same length I. By means of (35)
Fio. 6.
it is then easy to calculate the phase difference AF between the rays
1 and 2 which is due to the motion of the apparatus in the ether. For the
time t 1 which the ray 1 takes to travel from P to S t and back again we
first obtain
I
I
21
c v
, V 2 . <2)
c-\-v c I tr/cr
since e' on the way forth and back is parallel to the vector v. In the same
way we obtain the time t 2 which ray 2 takes to travel from P to S 2 and
back and, since in this case e' is perpendicular to v along the whole path,
we get by means of (35)
/ 2 2/(c 2 v 2 )~*. (63)
Neglecting terms of order higher than the second, we therefore get for the
mentioned phase difference
When the apparatus is rotated through an angle of 90, so that the path
PS 2 now becomes parallel to v, the difference in phase will be A-F.
Such a rotation of the apparatus should therefore cause a shift of the
interference fringes corresponding to a change in phase of 2 A F.
When the distance between the interference fringes is used as unit of
length, the phase difference 2A-F gives directly the shift of the inter-
ference fringes by such a rotation of the apparatus. In Michelson's
28 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 12
experiment, 2AF was about , so that one would expect a shift in the
position of the fringes of about of the distance between the fringes.
Despite the fact that Michelson would have been able to detect with
certainty a shift a hundred times smaller, he could not find any effect
at all. Thus for the first time we are confronted with an experiment
indicating that the principle of relativity is true with an accuracy
of at least second order, f
13. The contraction hypothesis
The result of Michelson 's experiment meant a very serious difficulty
for the ether hypothesis. Michelson himself tried to explain the absence
of the effect by assuming that the ether was carried along by the earth
during its motion round the sun. In this case there would be no ether wind
at the surface of the earth, but perhaps at high altitudes. Michelson
therefore repeated his experiment on a high mountain, but still without
any detectable effect. The assumption that the ether should be com-
pletely dragged along by the earth is also in conflict with all optical
experience and with Lorentz's electron theory, according to which the
dragging is only partial inside the refracting media.
In order to explain the absence of any effect due to the motion of the
earth in Michelson 's experiment, LorentzJ and FitzGerald indepen-
dently put forward the hypothesis that any rigid body moving with
velocity v is contracted in its direction of motion, the relative contraction
being equal to (1 v 2 /c 2 )*. The length of the path PS l in Michelson 's
experiment (Fig. 6) would then not be I, but 1(1 v*/c 2 )*, while the
length of PS 2 is unchanged, since PS 2 is at right angles to the direction
of motion of the apparatus. For the time t l we then obtain, instead of (62),
*! = 2/(c 2 -V)-i = t 29 (65)
t 2 being given by (63). In this case the phase difference &F becomes zero
in agreement with Michelson 's experiment.
According to this strange hypothesis, a stick which has the length 1
when it is perpendicular to the direction of motion of the earth should
obtain the shorter length
f Michelsori'.s results have boon confirmed later by several investigators. See, for
example, K. J. Kennedy, Ptoc. Nat. Acad. 12, 621 (1926), and K. K. Illmgworth, Phys.
Rev. 30, 092 (1927). Contrary to these results Miller obtained a small effect. See
D. C. Miller, Rev. Mod. Phys. 5, 203 (1933).
t H. A. Loreritz, Amst. Verh., Akad. v. Wet. 1, 74 (1892).
G. F. FitzGerald, see O. Lodge, London Transact. (A) 184, 727 (1893), in particular
p. 749.
I, 13 HISTORICAL SURVEY 29
when it is turned so as to be parallel to the direction of motion of the
earth. Certainly on the earth it will never be possible directly to measure
this shortening, since all bodies, thus also the measuring sticks, are
shortened at equal rates. An observer at rest in the ether outside the
earth would, however, in principle be able to observe the shortening
and he would find the earth and all objects on the earth contracted in
the direction of motion of the earth.
The contraction hypothesis looks rather startling at first sight, but, as
stressed by Lorentz, f it is impossible to escape from it as long as the
conception of an absolute unmovable ether is maintained. For, from
this view-point, the result of Michelson's experiment can directly be
taken as a proof of the contraction with the same justification as the
shift of the interference fringes when certain parts of the apparatus are
heated is taken as a proof of a change in length of the heated parts.
In order to make the hypothesis somewhat more acceptable, Lorentz
made an attempt at explaining the contraction phenomenon on the basis
of the electron theory. He actually succeeded in giving a plausible
explanation of the formula (66). Assuming that the material bodies
are built up of electrical particles which are held together exclusively
by means of electric forces, he was able to show that the equili-
brium positions of the electrical particles in such a purely electrical
system are changed in agreement with (66), when the system as a whole
is given a constant velocity in the ether. The difficulty was only that the
presupposition that the particles are held together exclusively by electric
forces could scarcely be assumed to be satisfied in the real substances.
In particular it was difficult to imagine how the charge of a single electron
could be held together, unless strong attractive forces of non-electrical
nature were active inside the electron. If one therefore assumes that the
contraction formula (66) is valid also for a single electron, as was actually
assumed by Lorentz, this must be regarded as a pure hypothesis which
cannot be based on the principles of the electron theory alone. The
Lorentz contraction therefore seemed to be a basic and universal pheno-
menon underlying the general laws of nature.
14. Validity of the principle of relativity for all physical pheno-
mena
Michelson's experiment was only the first of a long series of attempts
to determine the motion of the earth relative to the ether. These experi-
ments include both optical and purely electromagnetic arrangements,
t See ref., p. 21.
30 FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY I, 14
and in each case the result was completely negative. All phenomena
appeared to be independent of the motion of the earth. At the end it was
impossible to doubt that the principle of relativity is valid exactly, not only
for the mechanical phenomena, but also for all optical and electro-
magnetic phenomena.
We shall not here enter into a detailed discussion of all these experi-
ments, but confine ourselves to recalling Ives's experiment mentioned
in 5, which showed that the Doppler effect also to a second approxi-
mation depends only on the relative velocity of the source of light relative
to the observer, in agreement with the principle of relativity. This
fact as well as the results of some of the other experiments mentioned
above cannot be explained on the basis of the ether theory, even if the
contraction hypothesis is added to it; for the formula (22) does not
contain any quantity which has the dimension of a length. Consequently
a hypothesis regarding the contraction of distances alone will not change
the formula (22) into the formula (II. 90) verified by Ives's experiment.
Next, Lorentzf investigated the problem of which hypotheses must be
introduced beside the contraction hypothesis so as to make all predictions
of the ether theory in accordance with the principle of relativity now
verified by the experiments. He found that it was necessary in every
inertial system to use a special time, the so-called local time, which
is different from the time in the absolute ether system. According to
the contraction hypothesis, the length of a metre stick depends on the
absolute velocity of the inertial system considered. Similarly, the rate of
clocks and therefore also the unit of time should, according to this new
hypothesis, depend on the state of motion of the inertia! system. If the
basic equations of the electron theory in a moving system of inertia
are written in terms of these local time and space variables, they assume
the same form in any system of inertia. All electromagnetic phenomena
should therefore appear to be independent of the state of motion of the
frame of reference. In this way it was for some time made possible to
maintain the concept of an absolute ether by the introduction of new
hypotheses, until Einstein (1905) realized that the very foundations
of the ether theory were seriously shaken by the results of the above-
mentioned experiments.
t See rof., p. 21.
II
RELATIVISTIC KINEMATICS
15. Simultaneity of events
THE fruitless attempts to find any influence of the motion of the earth on
mechanical, optical, and electromagnetic phenomena gave rise to the
conviction among physicists that the principle of relativity was valid
for all physical phenomena. This obviously changes the whole basis of
our description of nature, for, as already mentioned in 2, the concept
of an absolute ether system loses its physical meaning as soon as the
universal validity of the principle of relativity is accepted. All physical
phenomena will then take the same course of development in any system
of inertia and one can never by any physical experiment decide which
system is the absolute one . All systems of inertia then become completely
equivalent and it must be required of a satisfactory theory that all
systems of inertia are treated on the same footing.
Einstein was the first to formulate this new standpoint and to draw the
consequences of it in his fundamental paper of 1905.f We have already
mentioned one of these consequences in 2. Since the fundamental equa-
tions of electrodynamics Maxwell's equations must hold now in any
system of inertia, it follows that the velocity of propagation of light in
vacuo must have the same constant value c 3 X 10 10 cm. /sec. in every
system of inertia. This is, of course, in conflict with the kinematical
concepts derived from our usual experience and which are expressed in
the addition theorem (I. 3). Thus the new experiences gained from
extremely accurate experiments, which are expressed in the principle
of relativity, compel us to make a revision of these kinematical concepts
which by habit had obtained an a priori validity in the minds of physi-
cists. Now Einstein could show that a closer analysis of the concept of
velocity, i e. a discussion of the methods by which a measurement of
velocities can actually be performed, opens up the possibility of an un-
ambiguous description of physical phenomena in accordance with the
principle of relativity.
Let us consider a light signal which travels in a straight line from a
point A to another point B in a given system of inertia. The velocity
with which the light signal has travelled from A to B is then defined as
the ratio between the distance from A to B and the time which the light
t A. Einstein, Ann. d. Phys. 17, 891 (1905); Jahrb. d. Radioaktivitat und Elektronik,
4, 411 (1907).
32 RELATIVISTIC KINEMATICS II, 15
needs to travel from A to B. The measurement of the distance does not
involve any difficulties, but on second thoughts it becomes clear that
the measurement of the difference in time between the emission of the
light signal from A and its arrival at B is not so simple. If we imagine the
time of emission ^ to be read on a clock placed at A, while the time of
arrival / 2 is read on another clock at B, the difference 2 ^ obtained in
this way will only give the real time which the light has taken to travel
from A to B if the clocks at A and B are put right. This obviously
requires that the hands of both clocks simultaneously are in the same
position. But how can we make sure that two events occurring in two
different places are simultaneous ?
It is possible to think of various methods to synchronize the clocks at
A and B. We can, for example, carry a third clock, which is set according
to the clock at A, from A to B and adjust the clock at B according to it,
or we can use a time signal which is sent from A to B. We shall start by
a consideration of the latter method which, in practice, has proved to be
the most accurate. Let us imagine that the time signal is emitted from A
when the clock at A records zero. In everyday life one would then usually
set the clock at B to zero when the time signal arrives at B. This is, of
course, not quite correct since the time signal is propagated with a finite
velocity. If we wish to be quite accurate, the clock has on the arrival of
the time signal at B to be put to l/u, where I is the distance from A to B,
and u is the velocity of the time signal. In order to be able to make this
correction it is thus necessary to know the velocity of the time signal,
but a measurement of a velocity presupposes, as shown above, that two
clocks in different places are synchronized, which was just the problem
which the time signal should help us to solve. Similar observations are
true when we set the clocks at A and B by transporting a third clock from
A to B. In this case, the transported clock has to be corrected for the
influence which the transportation might possibly have had on the
motion of the clock, but to find experimentally this influence it is
obviously necessary in advance to dispose of two clocks in different
places of which we know that they are synchronized. Here again we are
moving in a circle.
All methods for the regulation of clocks meet with the same funda-
mental difficulty. The concept of simultaneity between two events in
different places obviously has no exact objective meaning at all, since we
cannot give any experimental method by which this simultaneity could
be ascertained. The same is therefore true also for the concept of velocity .
As stressed by Einstein, we must first define what we understand by
II, 15 RELATIVISTIC KINEMATICS 33
simultaneity. As to definitions of concepts we are, however, to some
extent free and, as we shall see in the following, it is possible to use such a
definition of simultaneity that the velocity of light is constantly equal to
c in all inertial systems.
16. The relativity of simultaneity
Let us imagine that we are placed in an arbitrary inertial system /
and that we are provided with a large number of clocks (standard clocks)
which show the same rate when placed at rest at the same place. These
clocks shall be distributed at all places in / where we want to make time
measurements. In order to synchronize the clocks we shall use light
signals since we know experimentally a good deal about the propagation
of light. Fizeau's experiment and similar more accurate measurements
show, for instance, that the time which a light ray takes to traverse a
closed polygon is equal to the ratio between the total length of the
polygon and the universal constant c occurring in Maxwell's equations.
This time can be measured by a single clock placed at a fixed point of the
polygon, independently of the definition of simultaneity, and all distances
may be measured by means of standard measuring-sticks at rest in /.
Now we choose an arbitrary point as regulating centre, and in
order to synchronize the clocks at different places in 1, a light signal is
emitted from O in all directions. Let the clock at at the start of the
signal record the time ; when this signal arrives at an arbitrary point P
the clock is put to -f Z /c, where / is the distance from O to P measured
with standard measuring-sticks at rest in 1 . In this way all the clocks in
the inertial system / are set in a definite way. Two events occurring at
two arbitrary points P and P l arc said to be simultaneous when the
clocks at P and P l record the same time at the moment when the events
occur. Such a definition of simultaneity is completely justified if it can
be shown that it does not contain any 'inconsistencies. In this con-
nexion two condition;-* must be satisiied, viz.
1. A signal starting from O r seconds later than the regulation signal,
i.e. at the time t -\~r, shall arrive at P r seconds later, i.e. when the
clock at P records the time +^o/ c ~l~ T - This condition means that
the method of regulation of the clocks shall be independent of the
time when the regulation is made.
2. The method shall be independent of the choice of the point which is
taken as regulating centre.
The first condition is no doubt fulfilled, since all points in an inertial
system are equivalent, so that two standard clocks which have the same
3595.60
34 RELATIVISTIC KINEMATICS II, 16
rate when placed together at will also have the same rate when they
are installed at different points and P.
As regards condition 2, we have only to show that a light signal emitted
from an arbitrary point P l9 when the clock there records t l9 will arrive at
another arbitrary point P when the clock at this point records the time
t = tt+l/c, (I)
where I is the distance between P l and P. In order to prove the equation
(1) we may, for simplicity, assume that the time of departure ^ of the
signal coincides with the arrival at P l of the regulation signal from 0, i.e.
If the signal from P x immediately after its arrival at P is sent on to
the point 0, its time of arrival 2 at is, according to the experimental
results mentioned above (Fizeau's experiment), given by
since the light signal has actually traversed a triangular path of total
length Z x +Z-Ho ( see Fi g- 7 )
The clock at P showed the time t when the signal from P x arrived at P.
When the regulation signal from O arrived at P, the clock was put to
t Q -\-l Ql f c. If this signal had been reflected back to it would, according
to Fizeau's experiment, have arrived there when the clock at recorded
the time / +2/ /c. Now the signal from P l is, however, first sent down to
r J (o+^o/ c ) seconds later. According to the above assumption 1
it therefore arrives at when the clock there records the time
. (4)
II, 16 RELATIVISTIC KINEMATICS 35
This time is identical with the time t 2 given in (3). From (3), (4), and (2)
we now obtain
This is just the equation (1) which we wanted to prove.
Thus the regulation method used is independent of the point chosen as
regulating centre and in this way we have established a definite way of
recording successive events in the inertial system /. Any event occurring
at a point P at the moment when the standard ckrck at this place shows
the time t is simply said to occur at the time t. Then also the concept of
velocity assumes an exact meaning and, especially for the velocity of
light, we obviously obtain the value c in all directions. For a light signal
emitted from an arbitrary point P x at an arbitrary time ^ has just been
shown to arrive at another arbitrary point P at the time ^-fZ/c, where I
is the distance between P and P v
Let us now consider another arbitrary inertial system /'. Suppose that
in this system also we place a great number of standard clocks of identical
construction to those used in /. These clocks are distributed at the differ-
ent points of /' and regulated in the same way as the clocks in / by means
of light signals emitted from an arbitrary point 0' in I'. All distances in
/' are now supposed to be measured with standard measuring-sticks at
rest in /'. The measuring-sticks shall be of the same type as those used
in /, which means that they have the same length when brought to rest
relative to each other. Since, according to the principle of "relativity,
Fizeau's experiment gives the same result in /' as in /, it is clear that this
method of synchronizing the clocks in /' provides a consistent time
description.
When the distances and the time differences are measured with clocks
and measuring-sticks in /', it then follows in the same way as in / that
the velocity of light relative to /' is also constant and equal to c in all
directions. An event occurring at a given point P' of /' at the moment
when the clock at this point registers the time t' is said to occur at the
time t 1 relative to /'. In general this time will be different from the time
.at which the same event occurs relative to /. Two events occurring at
different points are now, of course, called simultaneous relative to /'
if they occur at the same time t r in /'.
Thus the concept of simultaneity has lost its absolute meaning since
two events occurring simultaneously for observers in / generally will
not be simultaneous for observers in /'. Let us, for instance, consider
two events occurring at two points A and B which are fixed in I. Since
the velocity of light is c in .all directions, the criterion for these events to
36 RELATIVISTIC KINEMATICS II, 16
be simultaneous relative to / is obviously that two light signals emitted
from A and B at the moment when the events occur shall meet in the
centre C of the line connecting A and B. A similar criterion for simul-
taneity is true also relative to /'. Now let the two events be simultaneous
relative to / and let us, for instance, imagine that the line connecting
A and B is parallel to the direction of the velocity v of /' relative to /.
Then consider the two points A 1 and B' in /' which, at the moment when
the events occur, coincide with the points A and B. Simultaneously
(relative to /) the centre C' between A 1 and B f will coincide with C.
Since now C', just as A' and B', moves together with 1' with a velocity
v relative to /, C' will not coincide with C at the moment when the
light signals from A and B meet in C. The light signals will thus not
meet in C' and, according to the above-mentioned criterion, the two
events are not simultaneous relative to 1' .
The concept of simultaneity between two events in different space
points consequently has an exact meaning only in relation to a given
inertial system. Only in the approximation where the velocity of light
can be regarded as infinitely great compared with all other velocities
which occur is it permissible to speak of an absolute simultaneity inde-
pendent of the states of motion of the observers. Such an approximation
is quite sufficient in daily life and in many cases also in physics, and this
explains the deep-rooted subjective belief in the existence of an absolute
time and in absolute simultaneity.
17. The special Lorentz transformation
In a given inertial system / an event which occurs at a point P at the
time t can be characterized by four figures, viz. the three coordinates
specifying the point P and the time parameter t. These four figures are
called the space-time coordinates of the event. If, for instance, we use a
Cartesian system of coordinates in the inertial system 7, the space-time
coordinates of the event are {x, y, z, t}, where x = (x, y, z) are the Cartesian
coordinates of the point P. The coordinates (x, y, z) are found by measur-
ing the lengths of the projections of the coordinate vector x on the
Cartesian axes by means of standard measuring-sticks at rest in the
inertial system /, while the time t is read on the standard clock which is
placed at rest at the point P.
In this way a definite system of space-time coordinates S is attached
to the inertial system /. When S is given, the frame of reference, i.e. the
inertial system /, is completely determined. On the other hand, different
systems of space-time coordinates S can, of course, be used in the same
II, 17
RELATIVISTIC KINEMATICS
37
frame of reference /, for example, we may use polar coordinates instead
of Cartesian coordinates for the specification of the different points in
space. In the special theory of relativity we shall, however, always use
space-time coordinates of the above-mentioned kind, and thus need
not distinguish between the frame? of reference / and the coordinate
system 8. On the other hand, in the general theory of relativity it will
appear necessary to differentiate between the frame of reference and the
system of coordinates which is used for the fixation of the events occur-
ring in the frame of reference.
s s'
V
x
O'
FIG. 8.
If we consider another inertial system, an event will also in this system
be specified by four space-time coordinates (x' , y', z f , t') defining a space-
time system of coordinates S' . The coordinates (jc',y', z', t') are found in
the same way as the coordinates in ti by means of standard measuring-
sticks and standard clocks now at rest in /'. Our primary task will be to
find the connexion between the space-time coordinates of the same
event in S and $', i.e. the transformation corresponding to the Galilean
transformation (I. 1) in non-relativistic kinematics. Since any uniform
translatory motion relative to S is uniform also relative to S', the
variables (x' ,y' \z' ,2') must obviously be linear functions of (x,y,z,t).
For convenience we shall assume that the Cartesian axes in S and 8'
are parallel to each other and that S r is moving relative to S with
velocity v in the direction of the positive #-axis. Let us, moreover, assume
that the origin Q' of S' coincides with the origin of 8 at the time
t = V - 0.
38 RELATIVISTIC KINEMATICS H, f 17
Now consider all the points in S' which form a plane
y' = a' = constant (5)
parallel to the z'z'-plane. These points will also form a plane
y = a = constant (6)
in S parallel to the #z-plane. The constants a' and a denote the dis-
tances between these planes and the #z-plane, and, since these distances
are measured by means of measuring-sticks in different states of motion,
the ratio K = a' fa (7)
might turn out to be different from 1.
The ratio K can, however, only depend on the relative velocity v and
a simple relativity argument shows that K actually must be equal to 1 .
For, if we change the signs on the x- and z-axes as well as on the x'- and
z'-axes, neither a nor a' is changed, but now the two inertial systems have
exchanged roles, S now moving with the velocity v relative to S' in the
direction of the positive #'-axis. Hence, we may conclude, just as before,
that / / I Q \
K a/a . (8)
From (7) and (8) it follows, however, that /c 2 = 1 and, since the positive
directions of the y- and the y'-axes are the same, a and a' must have the
same sign. Hence, K = ^ fl , = a (9)
These considerations show that an event occurring at a point with the
coordinate y in S will have a coordinate y' with respect to $', which is
given by V = y. do)
In the same way we find that the z-coordinate is transformed according
to the equation , __ n , v
z z. \ Li )
In order to find the transformation equations for the two other space -
time coordinates we make use of the fact that a light signal in S and in
S' is propagated in all directions with the same velocity c. If the light
signal starts frqjn the coinciding points and 0' at the time t = t' = 0,
the propagation of the spherical light wave is described in S by the
equation x * +y * +z *- c *t* = . (12)
In S' the wave is analogously given by the equation
s'2 +y '2 + 3'a_ c ya = o. (12')
Now put s 2 = # 2 +2/ 2 +z 2 -c 2 J 2 (13)
and *' 2 = * /2 +7/' 2 +z /2 -cV 2 . (13')
II, 17 RELATIVISTIC KINEMATICS 39
For any set of values of the variables (x, y, z, t) which makes s 2 equal to
zero, s' 2 must then also be zero, and since the connexion between
(z',y',z',O and (x,y,z,t)
is linear, this is only possible when s' 2 is proportional to s 2 , i.e.
s' 2 = K (v)s 2 , (14)
where K is a constant which can depend only on the relative velocity v.
By the same relativity argument which was used in connexion with
equation (8) we see at once that the constant K must actually be equal to
1. Thus (14) reduces to , 2 __ > nr;\
S S", V lc V
i.e. the quantity s 2 defined by (13) is an invariant.
By means of (10) and (11), (15) can be written in the form
x*-c*t* = x' 2 -c 2 t' 2 . (16)
Since (10) should be fulfilled for an arbitrary event, x' and t' must be
linear functions of x and t alone. Therefore we can write
x' = otx+8t,
(17)
t' -
where the constants , ]8, y, 8 have to be determined so that (16) is satis-
fied for all x and t.
For the origin O' we have x' = 0. Thus we obtain from the first
equation (17) for the motion of 0' relative to S
x - _#/,
and, since the velocity of 0' relative to S is v, it follows that
J8 - at?. (18)
For the origin we have, however, x = 0. By introducing x -- into
the equations (17) we get, after elimination of t, the equation
x' - j8/'/8
describing the motion of relative to S'. For symmetry reasons the
velocity of relative to S' must, however, be v, wiich together with
(18) gives ft = _ 8v = _^ ieg==a (19)
By means of (18) and (19) equation (17) can be written in the form
x' = oi(xvt),
(20)
f = yx + ott.
Introduction of these equations into (16) gives
(21)
40 RELATIVISTIC KINEMATICS II, 17
Since this latter equation will be satified by all possible values of the
independent variables x and t, the coefficient of the variables x 2 , t 2 , and
xt, respectively, on both sides of the equation must be equal. This gives
three equations for the determination of the two quantities a and y. The
last two of these equations give immediately
= (l-v 2 /c 2 )-* (22)
and ^-^ /C2= ^VO=5^- (23)
The remaining equation, expressing the fact that the coefficients of x 2
on both sides of the equation are equal, is then identically fulfilled, which
shows that equation (19), i.e. the assumption that relative to >S Y/ moves
with a velocity v, is in accordance with equation (16).
From (10), (11), (20), (22), and (23) we finally get the following trans-
formation equations for the space-time coordinates of an arbitrary event
, x vt ,
x = y =
, t vxfc 2
(24)
The inverse relations which are obtained by solving the four equations
(24) with respect to the variables x, y, z, t are
x'+vt'
-
-
They may be obtained from (24) by interchanging the primed and the
unprimed variables and replacing v by v.
In ( 1 8) the quantity v was introduced as the velocity of 0' relative to S.
It follows from (24), however, that any fixed point P' in S f with constant
values of coordinates x', y' , z' moves with velocity v relative to S in the
direction of the #-axis. Analogously, we see from (24') that each fixed
point P in S moves with velocity v relative to AS" in the direction of
the '-axis. The quantity v therefore denotes simply the relative velocity
of the two systems of inertia.
Lorentz was the first to introduce the transformation equations (24)
and (24') and, therefore, they are usually called Lorentz transforma-
tions. The derivation of these equations from the point of view of the
II, 17 RELATIVISTIC KINEMATICS 41
principle of relativity is, however, due to Einstein. | In view of the special
position of the Cartesian axes in Fig. 8, we here speak of especial Lorentz
transformation. Under such a transformation the quantity s 2 defined
by (13) is invariant. If we put c = oo, (24) goes over into the special
Galilean transformation (I. 2).
18. The most general Lorentz transformation
The transformation of the space-time coordinates in the more general
case, where the relative velocity of S' and S is not parallel to the
#-axis and where the rectangular coordinates in S and S' have arbitrary
orientations relative to each other, can obviously be obtained by means
of a suitable combination of spatial rotations of the axes in S' and 8
together with a special Lorentz transformation (24). Since s 2 is un-
changed by spatial rotations, s 2 is thus also invariant by these. more
general Lorentz transformations.
Sometimes we shall need explicit expressions for the Lorentz trans-
formations in the general case and it is convenient then to use the follow-
ing vector representation. Let us, for a moment, again consider a special
Lorentz transformation corresponding to the orientation of the coordi-
nates shown in Fig. 8. We can now depict the space vectors
x (x,y,z) and x' (x',y',z')
of the space points of the event considered in S and $', respectively,
in one and the same vector space, (x, y, z) and (#', y' ', z') being regarded as
the components of the image vectors in & fixed system of coordinates in
this abstract three-dimensional vector space. In the same vector space
the vector v, representing the velocity of the system S' relative to S,
is depicted as a vector with components (v, 0, 0). The special Lorentz
transformation (24) can then obviously be written as a relation between
the image vectors x, x', v and the variables t and t' of the form
where (v. x) = v x x-{-v v y-\-v z z\^ the scalar product of the image vectors
x and v. It is immediately seen that the three components of the vector
equation in (25) are identical with the first three equations (24).
Introducing in the same manner the image vector v' = ( v, 0, 0)
which represents the velocity of the systenr$ relative to S' ', the inverse
t Cf., for example, A. Emstoin, Ober die spezielle und die allgemeine Relatwitatstheome,
Braunschweig 1917.
42 RELATIVISTIC KINEMATICS II, 18
equations (24') can obviously be written in the form
X ~-=
. (25')
Since v' - -v (26)
the inverse equations (25') are in this case obtained from (25) by inter-
changing (x',') and (x,) and replacing v by v.
By a rotation of the Cartesian axes in S the components of x and v
undergo an orthogonal transformation, and since the axes in the abstract
vector space are supposed to be fixed, this means that the rotation of the
( 'artesian axes in S induces a corresponding inverse rotation of the image
vectors x and v, while the vectors x' and v' remain unchanged. In a
similar way, a rotation of the Cartesian axes in S' induces the corre-
sponding inverse rotation of the image vectors x' and v'.
Now let us first consider the case where the Cartesian axes in S and S'
are subjected to the same rotations (starting from their position as shown
in Fig. 8). This means that the sets of variables (x,y,z) and (x r ,y',z')
are subjected to orthogonal transformations with the same coefficients.
Then the image vectors x, v, x', v' also suffer the same rotations and
therefore the relations between these image vectors will still be given
by (25), (25'), and (26). In this case we speak of a Lorentz transforma-
tion without rotation, since the angles (measured in $and S', respectively)
through which the Cartesian axes in S and S' should be turned in order
to obtain the orientation shown in Fig. 8 are the same, so that in a certain
sense the Cartesian axes in S and S' have the same orientation (cf.,
however, the considerations in 19).
If ?' r , v ir r s denote the components of the velocity of the system S'
relative to ft, and if we write y = (1 v 2 /c 2 )-*, the vector equations (25)
are only a short way of writing the four equations
, (27)
-l)t^ ^
y' = (y-i)v'*aA>H-{i + (y-W
z f (y ]) VzVx x/v 2 +(yl)v s v y y/v 2 +{l + (y l)vl/v*}z v z yt
t r = - yv x x/c 2 yv v y/c z -~yv z z/C 2 -\-yt
which thus represent a general Lorentz transformation without rotation.
On account of (26) the inverse equations (25') are again obtained from
(27) by interchanging the variables (x,y,z,t) and (x',y',z',t') and sub-
stituting ( v x , Vy, ~v z ) for (v x ,v y ,v s ), respectively.
II, 18 RELATIVISTIC KINEMATICS 43
Proceeding now to the consideration of the case where the Cartesian
axes in S and S' do not have the same orientation, we must keep in mind
that the Cartesian axes in S and S' must be subjected to different rotations
in order to attain the orientation of the axes shown in Fig. 8. While the
last equation (25) remains valid without change, the first equation has
to be replaced by
Dx' - x+v{(y-l)(x.v)/*; 2 - r *}> (28 o)
where D is the rotation operator which transforms the image vector x'
into the vector Dx' corresponding to a Lorentz transformation without
rotation. Thus X)" 1 represents the rotation of the Cartesian axes in S',
which would give these axes the same orientation (in the above-men-
tioned sense) as the axes in S. Instead of the equation (26) we have
Dv' = ~v, (29)
i.e. the components (v' x , v y , v z ) of the velocity of S relative to S' are in
this case not equal to (v x , v y , v z ). Multiplying (28 a) by the inverse
rotation operator D" 1 and applying (29), the Lorentz transformation
may in this general case be written in the form
X' = 3>-ix-v'{(y-l)(x.v)/-y*} ]
*' = y{*-(V.X)/C} /'
D can also be interpreted as the rotation which has to be applied to the
axes in S in order to obtain the same orientation of the axes in S and 8'.
The relations inverse to (286) are therefore
x = X>x'-v(-
)> (28')
= v }
which also can be proved directly by introducing (28') into the right-
hand side of (286).
It is easily seen that the equations (25) and (28) satisfy the equation
(15) which can be written in the vector form
(x.x) c 2 t 2 == (x'.x') cV 2 . (30)
If we put c = oo, (25) becomes the general Galilean transformation
(I. 1).
Until now we have assumed that the origins and 0' coincide at the
time t = t' = and, accordingly, the Lorentz transformations are
homogeneous transformations of the space-time coordinates. We shall
now abandon this assumption and consider a displacement of the origin
of the space and time coordinates in S'. This means that we have to
replace (x' 9 y',z',t r ) in (24), (25), and (28) by x'-x' Q) y'-y*, z'~4 *'-&
44 RELATIVISTIC KINEMATICS II, 18
respectively, where x' Q , y' Q , z () , t' Q are constants. By an inhomogeneous
Lorentz transformation of this type the quantity s 2 will not be invariant
any more. However, if we consider two events with the coordinates
r t , //!, 2^ /! and j\ 2 , 7/0, z 2 > ^2> respectively, the differences
As -= .^-.r,,, A// //r~//2>
between 1 he coordinates of the two events will also be transformed for
an inhomogeneous Lorentz transformation according to the equations
(25), (28) since the constants .rj,, y' Qj z' , t' (} will disappear when the differ-
ences are formed. Therefore the quantity A.s >2 , defined by the equation
A,s 2 - A.r 2 | A?/ 2 + A;: 2 ~ r 2 A/ 2 , (31)
will be invariant by an arbitrary mhomogeneous Lorentz transformation.
According to the principle of relativity all physical phenomena have
the same course ol development in all systems of inertia and it must be
required that, in the theoretical description of the phenomena, all inertial
systems are treated on the same footing, i e. the fundamental equations
of physics must have the same form in every inertial system. In other
words the fundamental equations must be form-invariant or covariant
under Lorentz transformations. This requirement, which is the formal
expression of the principle of relativity, has proved to be very useful in
the development of new theories.
As we shall see below, this requirement of form-invariance is auto-
matically fulfilled ior Maxwell's fundamental equations of electro-
dynamics in vacuo. On the other hand, Newton 's fundamental equations
of mechanics do not satisfy this requirement, since these equations, as
shown on pp. 3-4, are covariant under Galilean transformations.
Newtonian mechanics, therefore, is valid only in the approximation where
Lorentz transformations and (! all lean transformations can be regarded
as identical, i.e. where all the velocities occurring are small compared
with the velocity of light. But for all mechanical phenomena, where
velocities of the same order of magnitude as the velocity of light are
involved, the Newtonian equations must be replaced by Einstein's
rclativistic equations of mechanics which are covariant under Lorentz
transformations (cf. Chapter III).
19. Contraction of bodies in motion
From the Lorentz transformation (24) we can now draw certain con-
clusions regarding the intercomparison of measuring-sticks and clocks
in the systems 8 and 8'. Consider a measuring-rod which is at rest
relative to 8' and is placed parallel to the o;'-axis (Fig. 8). The end-points
II, 19 RELATIVISTIC KINEMATICS 45
of the rod therefore have constant coordinates x^ and x\ and the length
of the rod in S' (its rest length) is
70 ~/ ~'
i ^ 2 ~~ *!
According to the first equation (24) the motion of the two end-points
relative to S is given by the equations
Now it is natural to define the length I of the rod relative to S as the differ-
ence between simultaneous coordinate values of the end-points; by
simultaneity in this connexion we understand simultaneity relative to S.
From (32) we then obtain
I - x 2 (t)~^(t) - (xi-XiKI-vifc*)* = Z(l-t' 2 /c 2 )*, (33)
which is independent of t. On the other hand, since the systems S and S'
are completely equivalent, a metre stick at rest on the .r-axis of S which
has the length 1 in 8 will have a length / relative to /S', which again is
given by (33).
A metre stick which is placed perpendicular to the a:-axis will, however,
according to (24), have the same length in S as in 8' . We may therefore
quite generally say that a body which moves with a velocity v relative
to an arbitrary inertial system 8 is contracted in the direction of its
motion according to the equation (33), while the transverse dimensions
are independent of the motion of the body. If F is the rest volume of the
body, i.e. the volume measured in an inertial system following the body
in its motion, its volume V in 8 is thus given by
V -- V Q (}-v*/c^-. (34)
The equation (33) is identical with the Lorentz equation (I. 66).
However, as regards the physical interpretation, there is a difference in
principle between the two equations. In (I. 66) Z was the length of the
metre stick at rest in the ether while I was the length of the stick in
motion with a velocity v relative to the ether. Thus, according to the
original point of view of Lorentz, the metre stick is attributed an absolute
length which is independent of the state of motion of the observer. In
equation (33), however, P is the rest length of the metre stick, i.e. the
length measured in an inertial system following the stick, and I is
the length measured in an arbitrary inertial system relative to which the
stick has the velocity v. Therefore, according to relativistic conceptions,
the notion of the length of a stick has an unambiguous meaning only
in relation to a given inertial frame, this length being different for the
46 RELATIVISTIC KINEMATICS II, 19
different systems of inertia. This means, however, that the concept of
length has lost its absolute meaning. We can only speak of an absolute
length in the approximation where the velocity of light can be regarded
as infinitely large.
In spite of the fact that the concept of simultaneity enters into the
above definition of the length of a moving rod, equation (33), as pointed
out by Einstein, can in principle be verified by experiment without the
use of clocks. Let us consider two rods M l and M 2 with the same rest
length 1 moving along the #-axis in S with the velocities v and v,
respectively. Since the length of a stick according to (33) depends on
the square of the velocity only, M l and M 2 must have the same constant
length I relative to 8, which means that they must coincide at a certain
time t. In other words, the coincidences of the two ends of M l with the
ends of M 2 must be simultaneous events relative to S. Let these
coincidences occur at the points A and B in S. Then a subsequent
measurement of the distance A B with a standard metre stick in S gives
the magnitude of I.
Even if it is, of course, impossible to perform such an experiment in
practice with an accuracy sufficiently high to verify equation (33), this
consideration shows that the Lorentz contraction given by (33) is a real
effect observable in principle by experiment. It expresses, however, not
so much a quality of the moving stick itself as rather a reciprocal relation
between measuring-sticks in motion relative to each other. In this con-
nexion it is natural to ask for the cause of the contraction. According to
the principle of relativity, the answer must be that such a question is
just as delusive as if, after the discovery of the law of inertia, the question
were put why a body left to itself will continue to move straight forward
with uniform velocity. While such a question was well justified in
Aristotelian physics it must be rejected as meaningless after Galileo's
discovery. According to Galilean and Newtonian mechanics only the
deviations from uniform translatory motions require a cause.
While Lorentz attempted to explain the contraction phenomenon on
the basis of the electron theory, Einstein's deduction of equation (33),
based on the principle of relativity alone, shows that the contraction
phenomenon is of a much more fundamental character. Instead of
considering the contraction to be a phenomenon which has to be ex-
plained on the basis of an atomistic theory of material bodies, it should
rather be regarded as something elementary which cannot be traced back
to simpler phenomena. Actually it represents a requirement which must
be satisfied by any atomistic theory, viz. the requirement of covariance
II, 19 RFLATIVISTIC KINEMATICS 47
under Lorentz transformations. If, on the other hand, this requirement
is fulfilled, the contraction of a macroscopic body in motion can obviously
be deduced from the theory of the atomic structure of the body.
Before leaving this section we shall consider the somewhat more
general case where the connexion between space-time coordinates is
given by a Lorentz transformation without rotation. Let x' x and X 2 be
the coordinate vectors of two fixed points Pi and P 2 in the system of
coordinates S'. The straight line connecting these points represents a
fixed vector r' = x' 2 xi in S'. At the time t the points Pi and P 2 will
have coordinate vectors x l and X 2 in S which are obtained from the
first equation (25) by putting x = x x , x' = xi and x = X 2 , x' = x 2 ,
respectively. Then by subtraction of these equations we obtain
o/ovi i ") V / / o c? \
-v*c*)~* 1}- - -, (35)
' ) -.2 ' x '
where r = X 2 x x
is the vector connecting two simultaneous positions of Pi and P 2 in S.
If r = TH+TJL is decomposed into its components r\\ and r parallel and
perpendicular to v, respectively, and analogously r' = r[,+r'_L, (35) can
also be written
which shows that it is only the parallel component which suffers a
Lorentz contraction. The inverse relation to (35) is
r = r'+v^-^{(l v 2 /c*)* 1}, (35')
which can be verified by inserting (35') in the right-hand side of (35).
From (35') it follows that a fixed vector in S' parallel to the #'-axis is
in general not parallel to the #-axis as judged by an observer in S. Even
in the case of a Lorentz transformation without rotation the Cartesian
axes in S' will thus from the point of view of an observer in S generally
not be parallel to the axes in 8. Hence if we state that the Cartesian
axes have the same orientation in a Lorentz transformation without
rotation this should be understood in the way described on p. 42. For
two fixed vectors ri and r 2 in S' which satisfy the condition (ri . r 2 ) =
we can very well have (r l . r 2 ) ^ 0. Therefore the rectangular axes of
coordinates in S 1 will generally not even be perpendicular to each other
when looked upon from the system S. For this reason it was necessary to
depict the vectors x and x' in the neutral abstract vector space intro-
duced on p. 41.
48 RELATIVISTIC KINEMATICS II, 20
20. The retardation of moving clocks. The clock paradox
Now consider a standard clock C' which is placed at rest in S f at a point
on the x'-axis with the coordinate x' x\ (Fig. 8). When the clock C'
records the time /' == f lt the standard clock in S which C' is passing by
at that moment will record a time ^ given by the Lorentz trarisforma-
tion (24'), ti = Y(t ' 1+vx ' 1/c 2) (y = (1_ V 2/ C2) -J).
Somewhat later, when C" records the time t' = t' 2 , it coincides with an-
other clock in $ which records the time
By subtraction of these equations we obtain
(36)
i.e. a clock which is moving with the velocity v relative to S will be slow
compared with the clocks in S.
When we keep in mind that the systems S and S' are equivalent it is
obvious that a clock at rest in 8 similarly will lag behind the clocks
inS'.
Since the Lorentz transformation from which this result is deduced is
based upon the method of synchronizing the clocks in S and /S', dis-
cussed in 16, one could think that the retardation of moving clocks
described by (36) was only apparent. However, just as in the case of
the Lorentz contraction, it can be shown that equation (36) contains a
statement regarding the rate of moving clocks which in principle may be
verified by experiment. Consider two standard clocks C l and C 2 placed
together at the origin of an arbitrary inertial frame of reference. At the
time t = the clock C 2 is set in uniform motion along the #-axis with
the velocity v. At the time t = t p it has reached a point P on the x-axis
and, according to (36), it will record the time t p (lv 2 /c 2 )*. Immediately
after arriving at P, C 2 is sent back to with the velocity v. It arrives
at when the clock C l records the time t = ^ = 2t p . Since, accord-
ing to (36), the rate of C 2 is independent of the sign of v, C 2 will on its
arrival at O record the time t 2 ~ 2t p (l v 2 jc 2 )*. The clocks C^ and C 2 ,
which at the start of C 2 both showed the time zero, will thus, after the
return of C 2 , record times t t and t 2 , respectively, connected by the
e( l uation t z = (l-t^/c'K. (37)
The difference between t 2 and t l may now in principle be measured
directly by a comparison of the readings of C^ and C 2 before and after the
whole process.
II, 20 RELATIVISTIC KINEMATICS 49
As mentioned by Einstein in his first paper on the theory of relativity,
this consequence of the theory gives rise to a paradox which in the past
has played an important role in the discussions on the consistency ot the
theory. Suppose that we introduce a frame of reference E which follows
the clock C 2 during its motion from to P and back again. Since, now,
the motion of C relative to R is quite analogous to the motion of
C 2 relative to /S, one would think that an observer in R would find that
the clock C\ is slower than C z , in contradiction with (37). This argument
is wrong, however, since equation (36) is valid only in an inertial
system and therefore it is not applicable to the system R which during
the change of velocity of C\ from v to v is accelerated relative to the
fixed stars. The question which equation should be used in R instead of
(36) cannot be answered in the special theory of relativity which only
allows treatment of the physical phenomena in frames of reference in
uniform motion. This discussion clearly shows the desirability of an
extension of the special theory of relativity to a general theory which
allows the use of systems of coordinates in arbitrary motion. (For the
final solution of the clock paradox, see Chapter VIII, 98, p. 258 )
Now let us again consider a standard clock moving with a uniform
velocity u relative to an inertia! system #. The time recorded by the
moving clock itself is called the proper time r of the clock. According to
(36) we have the following relation between the increase in proper time
dr and the increase dt ol the time in S.
dr= (l-u*/c*)* dt. (38)
This equation is now assumed to be valid also for an arbitrarily moving
clock where u is the momentary velocity of the clock. Hence we assume
that the acceleration of the clock relative to an inertial system has no influence
on the rate of the clock, and that the increase in the proper time of the clock at
any lime is the same as that of the standard clocks in the rest system &, i.e.
the system in which the clock is momentarily at rest.
It should now also be possible to deduce the retardation of moving
clocks from the fundamental laws of mechanics governing the running
of the clockwork. But, just as in the case of the Lorentz contraction, it
is more adequate to regard the retardation phenomenon as an elementary
phenomenon which is a direct consequence of the principle of relativity.
If we took Newtonian mechanics as a basis for the calculation ot
the working of the clock, no retardation of moving clocks would be
found , since the time in Newtonian fundamental equations is an invariant
parameter (cf. I. 1 6), but this just shows that the Newtonian equations
3595.60
50 RELATIVISTIC KINEMATICS II, 20
are not accurate in the region where (1 v 2 /c 2 )* differs appreciably from
1. If, on the other hand, we use the exact relativistic equations of
mechanics in the description of the working of the clocks (cf. Chapters
III and VI), the retardation effect must of course follow as a consequence
of these equations.
In view of the fact that an arbitrary physical system can be used as a
clock, we see that any physical system w r hich is moving relative to a
system of inertia must have a slower course of development than the
same system at rest. Consider, for instance, a radioactive process. The
mean life r of the radioactive substance, when moving with a velocity v,
will thus be larger than the mean life r when the substance is at rest.
From (36) we obtain immediately
r = (1 ^' 2 /c 2 )~M. (39)
In general, v is so small relative to c that we need not discriminate
between r and r. In recent years, however, very rapidly moving radio-
active systems the mesons have been observed in cosmic radiation
where the factor (1 v 2 /c 2 )~ J can be of the order of magnitude of 100 or
more. Therefore equation (39) has been o( essential importance in the
interpretation of the phenomena connected with the decay of the mesons.
We may also use a radiating atom as clock, the number of light waves
emitted per second being a measure of the rate of the atomic clock. If
v is the proper frequency of the atom, i.e. the frequency of the emitted
light measured in the system of inertia /S in which the atom is at rest,
the number of waves emitted per unit time in this system is just v.
In the system S relative to which the atom is moving with the velocity v,
the number of waves emitted per unit time will then be v(l v*/c 2 )*, for,
according to (36), a unit time interval in S corresponds to the time
interval AT = (1 v 2 /c 2 )* in the rest system $. If now the moving atom
has no radial velocity relative to an observer in S, this number will be
equal to the frequency v found by the observer in S, the number of
emitted waves being equal to the number of waves arriving per time unit.
Consequently we have v = ^^^2^2)^0 ( 40 )
when the radial velocity *is zero. This is, for example, the case when
the atom moves with the velocity v along a circle whose centre is the
observer or when the direction of the light arriving at the observer is
perpendicular to the direction of motion of the atom.
According to the theory of relativity we must therefore expect a shift
in frequency also for perpendicularly incident light, viz. a shift towards
smaller frequencies, in contradistinction to the non-relativistic Doppler
II, 20 RELATIVISTIC KINEMATICS 51
formula (I. 14). This red shift of the spectral lines, the so-called 'trans-
verse' Doppler effect to which we shall come back later ( 25, p. 62), is
a direct consequence of the retardation of moving clocks described by
(36), and any experiment which permits an experimental verification
of this effect is therefore simultaneously an experimental proof of
formula (36).
21. Transformation of particle velocities
Let us again consider the two systems of inertia S and S' (Fig. 8) whose
space-time coordinates are connected by (24) and (24'). The motion of
an arbitrarily moving particle will then in S be described by a set of
equations x = ^ y = y(()> g = ^ (41)
In S' the same motion is described by functions
x' = x'(t'), y' = y'(t'), z' = z'(t'), (41')
which may be obtained from the functions (41) by means of the Lorentz
transformations (24) The momentary velocity of the particle relative
to S' is defined by
,
u . ,,- (42)
Analogously, the corresponding quantities in S are given by
Idx du dz\ \
- , , I I / A e\\
' dt' dt) . (43)
w = (? " '
Differentiation of the special Lorentz transformation (24) now gives
dx' y(dxvdt), dy' -^ dy, dz' = dz \ . .
} > v"* 4 */
dt' = y(dt-v dx/c 2 ) )
from which we immediately obtain
' ^x ^ ' / M'v\ I ^ r- 1 ) ' 'M'z\ * ^ /^ / / 4 -\
1l x = , 11 . = 2- , U z = ; . (45)
These equations for the transformation of the velocity reduce to the
ordinary transformation equations (I. 4) in the limit as c -> oo. The
relations inverse to (45) are obtained in the usual way by interchanging
the primed and the unprimed variables and substituting v for v.
52 RELATIVISTIC KINEMATICS II, 21
If we choose the 2-axis so that u (and thus also u') is perpendicular to
the z-axis, (45) can be written in the form
f Q f U COS I/ 1 V f Q f t*/ OXIJL i/ JL t/
1 UVCQS&/C 2 ' lUVCOSl
where i? and i? 1 ' are the angles between the #-axes and the vectors u and
u r , respectively.
From these equations we obtain directly
tan T?' = Sm ^-~- - , (46)
' - { 1 ~~ 2v CQS &lu ~ ^ v2 / u * ~~ 1?2 8ln2 #A' 2 i "
These equations are the relativistic generalization of the equations
(I. 5) and (I. 6). From (47) we obtain by an elementary calculation the
formula (J __.^/ c 2) (l _^/ c 2)i _ (i^ u 2/ c 2^i_ v 2f c ^} m (48)
If u, and thus also u' 3 are parallel to the #-axis, we get from (45) the
rHativistic addition theorem for velocities
' __ ^-~ ?; __ u'-\-v
For u' -- c it gives also u --- c.
From the Lorentz transformations it follows directly that no systems
of inertia >S y/ can exist for which v > c, since the equations (24) as well as
the expressions for the Lorentz contraction and the retardation of clocks
would become imaginary in this case. But it can be shown, furthermore,
that particles (or, more generally, signals) cannot move with a velocity
greater than c relative to any inertial system, since this would lead to
absurd results. Let us assume for a moment that we were able to emit
signals with a velocity greater than the velocity of light. At the time
t -- t' --- 0, where the two systems of coordinates 8 and 8' in Fig. 8
coincide, we could then send a signal from the common origins 0, 0'
along the negative u/-axis with a constant velocity u' > c relative to AS".
At the time t\ > 0, this- signal would arrive at a point P on the negative
a 1 ' -axis with the coordinate x' p ---- iit\. The space-time coordinates of
this event in 8 are, according to (24'),
Immediately after its arrival in P the signal is supposed to be sent back
II, 21 RELATIVISTIC KINEMATICS 53
to O with a velocity w > c relative to S. The motion of the signal is
thus described by the equation
x = w(t-tj+x p . (51)
This signal will arrive at the origin of S at a time t 2 which is obtained
from this equation by putting x = 0, thus at the time
/ 2 = t^Xpliv = yf^lu'v/ct+W v)/}. (52)
If we now choose u' and w so that
u' > c*/v, w ^ ~ -- (53)
1 u f v/c z -l v '
we could obtain that / 2 - ; 0, (54)
i.e. that, at the return of the signal to O, the clock at O records a number
which is smaller than that recorded by the same clock at the moment of
departure of the signal. Obviously this is impossible, and therefore we
can infer that in nature no signals can exist which move with a velocity
greater than the velocity of light relative to any system of inertia. This
represents a general statement regarding the fundamental laws of
nature. ]n particular, as we shall see later (Chap. Ill, 29, p. 76), a
material particle according to relativistic mechanics can never reach
velocities larger than r.
By differentiation of (25) and (25') we obtain the following transforma-
tion equations for velocities in the case of Lorentz transformations
without rotation
_
l-(v.u)/c a
(l_^ c 2)iuM-I-l-?;c
U =
These equations are reduced to (I. 3) in the limit as c -> oo.
From (55) we deduce
(1 V.U/c 2 )(L w'^ 2 )* - (1 v 2 /r 2 )i(l-M 2 /r, 2 )i (56)
in agreement with (48).
When u' is perpendicular to v, i.e. for u'.v 0, (55') reduces to
u = v + (l i; 2 /c 2 )iu' (57)
and when u is parallel to v we get back to the formula (49).
22. Successive Lorentz transformations. The Thomas pre-
cession
Let us now consider three incrtial systems S, S', and S" of which S'
moves with the velocity v relative to 8, while S" moves with the
54 RELATIVISTIC KINEMATICS II, 22
velocity u' relative to S'. The connexion between the coordinates
(x,t) in S and (x',f) in S' is then given by a (generally inhomo-
geneous) Lorentz transformation. In the same way, the connexion
between (x',f) and the coordinates (x" ,t") in S" is represented by a
Lorentz transformation. By elimination of the four variables (x',f)
between these eight equations we obtain the connexion between (x, t)
and (x", t") and, for physical reasons, this relation must also be a Lorentz
transformation. Mathematically this is expressed by the statement that
the Lorentz transformations form a group. If the origins in 8 and 8'
coincide at the time / ~~ t' 0, and if the same is the case with S' and
8" at the time t' t" -= 0, the origins in 8 and S" will of course coincide
at the time t t" - 0. This shows that the homogeneous Lorentz trans-
formations form a subgroup. It is also clear that the pure spatial rota-
tions of the Cartesian axes without any change of the system of reference
form a subgroup.
In non-relativistic kinematics the Galilean transformations without
rotation of the Cartesian axes also form a subgroup. This is, however,
not the case in relativistic kinematics, for if we combine two Lorentz
transformations without rotation the resultant Lorentz transformation
will in general correspond to a change of orientation of the Cartesian
axes. Let the transition from 8 to 8' be given by the Lorentz transforma-
tion (25), while the transition from 8' to 8" is described by the equation
obtained from (25) after replacing (x,,v) by (x',',u') and (x',') by
(x", t"). By elimination of x', t' we then obtain a Lorentz transformation
of the type of (286), viz.
X" - D-IX-W"(*^[ * 1"[ - -}, (58)
where the operator D in general is different from the unity operator.
w is the velocity of the system 8" relative to S and w" is the velocity of
S relative to S". Since the transformations from S to S' and from S' to
8" were Lorentz transformations without rotation, the velocity of S'
relative to S" is equal to u' while the velocity of 8 relative to S' is
equal to v. We now obtain the velocity w of S" relative to S from (55'),
identifying u with w in this formula, i.e.
w __
1 + (U'.V)/C* ' ''
From the same equation we get the velocity w" of S relative to S" by
II, 22 KELATIVISTIC KINEMATICS 55
replacing v and u' by u' and v, respectively; thus we have
_ _ - --
1 + (!'. V)/C ' ( '
According to (29) we have in our case
Dw" = ~w, (60)
and a comparison of (59) and (59') shows that the rotation operator D
in general is different from the unity operator. Only if u' is parallel to
v, say equal to kv, we get from (59) and (59')
l+k
w = ~ w = r v >
i.e. in this case ) 1 and the combined Lorentz transformation is also
without rotation.
Let us now consider the case where the transition from S' to S" is an
infinitesimal transformation, i.e. where u' is infinitesimal. Neglecting
all terms of higher than first order in u', the transformation from S' to
S" is reduced to
x" = x'-uY, t" - J'-(u'.x')/c 2 (61)
and for w and w" we obtain from (59) and (59')
W
,. = _/v + '-v<X'?l>'
(62)
By substituting the expressions (25) for x' and t' in (61) we obtain
an equation which after, a simple, but lengthy, calculation may be written
in the form (58) with
dv = w v
Thus, we have
Dx =
The rotation operator I) thus represents an infinitesimal rotation around
the direction of the vector ft, the angle of rotation being equal to the
magnitude |ft| of the vector ft. With I) given by (64) and w and w"
56 RELATIVISTIC KINEMATICS IT, 22
given by (62) it is easily verified that the equation (60) is satisfied to
the approximation considered.
Let us now consider a point compass, i e. a material particle which in
some way or other defines a direction. It may be assumed that a classical
electron with spin represents such a point compass. Tf the velocity of
the particle relative to 8 is v ~ v(/), and if w r e put dv \(t) dt in (64),
the systems 8' and 8" in the above-mentioned considerations will be
momentary rest systems of inertia for the particle at the times t and
t-\ dt, respectively. Since the transition from 8' to 8" represents an
infinitesimal Lorentz transformation without rotation, it is natural to
assume that the direction shown by the compass at the time 1 -\-dt has
the same orientation relative to the axes in 8" as it has at the time t
relative to the axes in 8' , provided that the forces on the particle do not
exert any torque on the compass.
Now, if we put dv - v dt, the rotation vector i defined by (64) repre-
sents the rotation which has to be applied to the axes in 8 at the time
t-}-dt in order to give them the same orientation as have the axes in #".
Since, furthermore, the direction of the compass relative to the rest
system is constant, this means that the direction of the compass relative
to 8 is turned through an angle corresponding to the rotation vector SI.
In other words, the compass performs a precession relative to 8 with the
velocity of precession
w = --^K l -^ 2 ) ! - l}(v X v), (65)
where v - r/v/rf/ is the acceleration of the point compass. When?' 'ewe
obtain to a first approximation
This precession phenomenon was studied for the first time by Thomasf
and is therefore called the Thomas precession.
23. Transformation of the characteristics of a wave according to
the theory of relativity
Let us again consider a plane wave with the wave normal n in the
.ry-plane of a system of coordinates 8 and with the frequency v and the
velocity w relative to 8. It is described by one or several wave functions
of the form / , . \
T =-, A<** m ,,-- TCOS ^Pl n J? , (67)
t L. W. Thomas, Phil May (7), 3, 1 (1927).
II, 23 RELATIVISTIC KINEMATICS 57
where a is the angle- between the wave normal n and the x-axis. In a
system of coordinates S' moving in the direction of the ^-axis with the
velocity v relative to S (cf. Fig. 1), the wave is described by corresponding
functions obtained from (67) when replacing the quantities occurring in
the phase by the corresponding quantities measured in the system of
coordinates S'. From exactly the same argument as that used in 3
it follows that the phase must be an invariant, i.e. the equation
(, # cos tx -)-?/ sin <x\ ,/,/ .^' cos cx' + ?/' sin a'\ , ^
t --- __.^ -- _-r v \t -- ^ -- (bS)
w I \ w I
must be valid for all points in space and at all times. Now using the
connexion between the space-time coordinates in A r and S' given by the
Lorentz transformations (24') we can eliminate the variables .r, //, t in
this equation and get
1 v cos OL/W , cos ex vw/c~ vx' v sin a ,
~~~ V ~~ ~ ~ ~ y
, isnrx , . .
----- y , (<>9)
w w
an equation which must be satisfied for all values of the independent
variables /', .r', //. This is possible only if the coefficients of /', .*;', y' ',
respectively, on both sides are equal. Therefore we must have
, I--VCO&OL/W 1 (v.n)/w
v sn (\ v sn <x
W W
v' cos OL i/(cos a rw/c 2 )
w' w(\ ?; 2 /c 2 )-
From these equations we obtain
(70)
.,
COS ex VWJC*
wvconoi
2vw cos oc v 2 w* v* sin 2 a
~~
(71)
. ^
The inverse relations are obtained in the usual way by interchanging
the primed and the unprimed quantities and replacing v by v. In the
limit as c -> oo, the equations (70), (71), and (72) reduce to the non-
relativistic formulae (I. 14), (I. 16), and (I. 19).
A comparison of the transformation equations (46) and (47) for the
58 KKLATIVISTIC KINEMATICS II, 23
velocity and direction of a material particle with the formulae (71) and
(72) shows that (46) and (47) become (71) and (72), respectively,
when we put u -- c 2 /w, u' -- c 2 /w'. In other words, the velocity u of a
particle and its direction n are transformed in the same manner as the
corresponding quantities for a wave with the phase velocity w c 2 /u
and direction n. In his wave theory of elementary particles de Broglief
made use of this circumstance by attributing to a particle with the
velocity ?/ and the direction of motion n a plane wave with the same
direction of propagation n and the phase velocity w c 2 /u, a procedure
which thus is relativistically invariant. If the particle velocity u = c
the corresponding wave velocity is w = c, which shows that direction
and velocity of such a particle are transformed in the same way as are
direction and velocity of a plane light wave in vacuum, but this only holds
for this particular value of the velocity.
24. The ray velocity in moving bodies
Consider a homogeneous isotropic medium, with the refractive index
??, at rest in the system of coordinates S' (Fig. 8). Relative to S the
medium moves with the velocity v. In the rest system S', Maxwell's
phenomenological equations of electrodynamics in dielectric bodies are
valid, and, according to the principle of relativity, this must be true for
any constant \ elocity of xS r/ relative to the fixed stars. The phase velocity
of light relative to $' is, therefore, w' c/n in all directions. In this
system of reference the ray velocity must, however, be equal to the phase
velocity, since the elementary waves which, according to Huyghens's
principle, determine the ray velocity are spherical waves with the con-
stant velocity , / /r . rt ,
J w --- c/n. (73)
This is not so, however, in the system of coordinates S. Let us con-
sider, for instance, the propagation of an elementary wave which, at the
time / -- t f - 0, is emitted from the common origin 0, O' of S and 8'.
In >S r/ the propagation of the wave is described by the equation
x' 2 +y' 2 +z'* w'H' 2 = 0, (74)
where w' is given by (73). By means of the Lorentz transformation (24)
we obtain from (74) the following equation for the propagation of the
elementary wave in S:
=
t L. do Brogho, Th&so, Pans 1924.
II, 24
RELATIVISTIC KINEMATICS
59
An elementary wave which is emitted at the time t from a point (:r , T/ O )
in the #?/-plane of the system S will thus, at the time J-f A, have a curve
of intersection with the o;?/-plane which is described by the equation
My; *o,yo) - (*-*o-a to)*/b+(y-y Q )*-bw'* A* 2 - 0. (76)
We shall here only deal with the case n > 1, i e. w/ < c. In this case
c > a > and 1 > 6 > and the curve of intersection consequently
y
X
\
FKS. 9.
is an ellipse with its centre at the point (x (} -\-a A, y Q ) and with the semi-
axes bw' A and b^w' A, respectively. The elementary waves are thus
dragged along by the medium with the velocity a, and simultaneously
the waves are contracted in the direction of motion in the propor-
tion b^.
Now let us again consider a plane wave with a normal n which lies in
the #?/-plane and makes an angle a. with the #-axis. The connexion
between a and the direction of the normal in 8' is then given by (71) or
by the inverse equation
tana
(l--?; 2 /r 2 )ismrx'
(77)
cos a' -\-vw' /c 2
Let cr in Fig. 9 be a wave plane in S whose line of intersection with the
xy -plane is given by the equation
X Q cos oc +2/ sin a ~ C = constant. (78)
The ellipse E with its centre at the point Q: (x Q -{-a A, y Q ) represents the
elementary wave at the time t-\-At, which is emitted from the point
P: (x , y ) at the time t. It is given by the equation (76). The wave plane
60 RELATIVISTIC KINEMATICS II, 24
at the time t-\ A is represented by the line or v which is obtained as the
envelope of the manifold of ellipses (76) which are obtained by varying
the parameters (x Q9 y Q ) in accordance with (78).
The direction and magnitude of the ray velocity are thus given by
the line PP V where P l with coordinates (#i,?/i) is the point at which E
touches the envelope a v We have obviously
_ y.
PP l ^ u A/, (79)
where u is the ray velocity. Since P l is the limiting point of intersection
of two adjacent ellipses of the manifold of curves (76), we see that the
coordinates of P l besides satisfying the equation (76) must satisfy the
equation obtained by varying :r , ?/ in (76) in accordance with (78), i.e. the
equation
- - sin a ---- cos ex
or (x XQ a A/)sina - b(y ?/ )cosa. (80)
From the three equations (76), (78), and (80) we obtain by elimination
of ,r and y Q the equation for the envelope <7 1 in the form
.rcostt-f-ysinrc ~- C \-\a-\- w'{b(b-\ tan 2 t\)] J ]costt Af. (81)
The last term in (81 ) gives the distance PA between the two wave planes
which must be equal to w A/, where w is the phase velocity. Therefore
we obtain
(82)
By inserting the expressions (75) and (77) for , 6, and a it is easily
verified that (82) is identical with the inverse of the equation (72).
According to (79) we have
11 X I~~ X Q __ V \-~lh /oo\
*-^Ar' ^-~AT~ J (83)
and, since r x lt y ~ y l is a solution of the equations (76) and (80), we
obtain by a simple calculation
fc* tan raw/
When inserting the expressions (75) and (77) for a, 6, a and remembering
that the ray velocity u' is equal to the phase velocity w' in $', we find
that (84) is reduced to
-- --'--
The equations (85) are identical with the relations inverse to (45), which
II, 24 KELATIVISTIC KINEMATICS 61
shows that the ray velocity (just as in the absolute ether theory, cf.
(I. 33)) is transformed in the same way as the velocity of a material
particle. When & and &' a.' denote the angles between the x-axes and
the direction of the ray measured in 8 and $', respectively, we have
therefore also for the ray velocities the equations (46) and (47) connecting
the quantities u, u' , &, and &' . By solving (47) with respect twit, we obtain
(86)
Here ^' ~ c/w, (87)
and e is a unit vector in the direction of the ray, i.e.
v.e i'cos#. (88)
In vacuum, i.e. for n - I, u' w r r, u and w are consequently
equal to c, and the formulae (46) and (47) for the ray velocity become
identical with the equations (71) and (72) for the phase velocity. Accord-
ing to the theory of relativity , the ray velocity in vacuum is thus identical
with the phase velocity in any inert ial system. In the absolute ether
theory this was the case only in the absolute system. This difference is
due to the fact that the elementary waves in vacuum according to the
theory of relativity are spherical waves with a fixed centre in any system
of inertia. (For w r ~ c we get from (75) a --- and b -- 1 .) We shall see
later (Chapters V, VII) that the ray velocity is equal to the velocity with
which the electromagnetic energy is flowing in an electromagnetic wa\ e.
The energy current density is, however, gi\ en by Poynting's vector, and
for a plane wa\ e in vacuum it is found that Poynting's vector lies in the
direction of the wave normal in all systems of inertia.
This identity of ray velocity and phase velocity applies only to the
vacuum. In a refractive medium we have in general to distinguish
between the two velocities; only when the ray is parallel to the direc-
tion of motion of the medium, both the ray velocity and the phase
velocity are, according to (85) or (86) and (72), given by the same formula,
ViZ " = = &*?., (89)
lv/cn ^ '
where the plus or the minus signs should be taken depending on whether
the ray travels in the same direction as the medium or opposite to it.
62 RELATIVISTIC KINEMATICS II, 25
25. The Doppler effect, the aberration of light, and the dragging
phenomenon according to the theory of relativity
The relativistic formula for the Doppler effect is obtained from (70)
by identifying the inertial systems 8 and 8' with the rest systems of
the observer and of the light source, respectively. The frequency v is
then the frequency measured by the observer, and v is equal to the
proper frequency v of the light source. Now, in vacuum, w = c and the
direction of the ray is equal to the direction of the wave normal. Thus,
we obtain l_Cv eWr
<> = LiirJ/r (90)
//I 2 //2\ \ /
/ v \ / /
where v is the velocity of the light source relative to the observer and
e = n gives the direction of the light in the system of the observer.
For (v. e) = 0, i e. when the direction of the observed light is perpendicu-
lar to the direction of motion of the light source, we get the formula for
the transverse Doppler effect v -- v(l v 2 /c 2 )*, which, as mentioned
in 20, is a direct expression for the retardation of moving clocks. As
mentioned in 5, the formula (90) has been experimentally verified with
high accuracy by Ives,f who measured the frequency of the light emitted
from rapidly moving ions.
The formula (46) connecting the directions of a light ray in two inertial
systems 8 and 8' leads immediately to the relativistic aberration
formula, when we take 8 and 8' to denote the systems of coordinates in
which the fixed stars and the earth, respectively, are at rest. At a point
P' outside the atmosphere of the earth we have u c and, when intro-
ducing the angles 6 # TT and 6' = &' IT between the direction of
motion of the earth and the actual and apparent directions to the star,
we obtain the relativistic aberration formula
COS0 + V/C
It deviates from the equation (I. 61) of the ether theory by quantities
of second order only, a deviation which is negligible compared with the
present accuracy of the measurements.
Since the atmosphere of the earth is at rest relative to 8', it follows at
once from the principle of relativity that the ray during its travel from
the point P' to the telescope does not undergo any further aberration.
According to the theory of relativity this should even hold exactly,
while Lorentz's absolute theory of electrons gave this result to a first
approximation only (cf. 10).
t See ref., p. 10.
II, 25 RELATIVISTIC KINEMATICS 63
Another essential difference in the relativistic treatment is that here
the aberration also appears in the direction of the phase velocity. While,
in the absolute theory, the wave normal according to (I. 28) has the same
direction in S and in S', the relativistic equation (71) for the transforma-
tion of the wave normal in vacuo is identical with the transformation
equation for the direction of the ray (46).
Neglecting all terms of higher than first order in v we obtain from equa-
tion (86) for the ray velocity in a medium moving with constant velocity
v relative to an arbitrary inertial system 8 the simple formula
u = u '+v.e(l u' 2 /c 2 ) =- c/r*+o:(v.e), (92)
where a 1 1/n* is Fresnel's dragging coefficient. Equation (92) is,
in this approximation, identical with Fresnel's equation (I. 48) for the
velocity of light in the 'absolute' system of coordinates. This formula
thus follows directly from the principle of relativity without any hypo-
theses regarding the atomic structure of the medium. It should be
stressed, however, that Fresnel's theory is true to a first approximation
only in the 'absolute' system. In any other system Fresnel's equations
are not valid even to a first approximation. According to the theory of
relativity we have, for example, u' - c/n in the rest system of the
medium whereas u' according to Fresnel's theory is given by the more
complicated formula (I. 46).
Equation (92) has been verified not only by Fizeau's experiments
discussed in 8, but also with high accuracy by Zeeman.f In these latter
experiments the velocity of light in a rapidly moving quartz rod was
measured. Indeed Zeeman's measurements were so accurate that it
was necessary to take into account the effect which, according to
Lorentz's electron theory, should occur in a dispersive medium (cf. 9).
Since n = n(v) in such substances is dependent on v, and since v on
account of the Doppler effect depends on the system of coordinates, it
must be specified which value for v, and thus for n, should be introduced
into equation (92). It follows from the derivation of (86) and (92) given
above that we have in (92) to use the value of n corresponding to the
frequency v in the rest system of the medium. When we neglect all quan-
tities of order higher than the first, the connexion between v and v,
according to (70), is given by
v'v -= -vn(v.e)lc, (93)
the difference between the directions of the ray and of the wave normal
t P. Zeernan, Amst. Versl. 23, 245 (1914); 24, 18 (1915).
64 RELATIVISTIC KINEMATICS II, 25
being of second order in v. Hence,
c c c dn nv(v . e)
n(v) n(v) n 2 dv c
where n = n(v) means the value of n for the frequency v. By using
this formula in (92) we obtain
n
(94)
This formula was in complete agreement with Zeeman's measurements.
The reflection of light by moving mirrors and the refraction of light at
the transition between moving media can be treated in a similar way
on the basis of the theory of relativity. In a system of inertia 8' ', in which
the mirror or the medium is at rest, the usual law for reflection of a light
ray by a mirror and the well-known law of refraction at the transition
between two media are valid. The corresponding laws in the system S
are then directly obtained by applying the transformation equations
(46) and (47) for the ray velocity. Experiments on reflection by moving
mirrors have been performed by Sagnacf and others. The result of these
experiments is in accordance with the theory.
In all these experiments, only effects of first order could be measured,
since the velocities with which the mirrors and the refractive media could
be moved were so small compared with the velocity of light that the terms
of second order in equation (86) were negligible compared with the
accuracy of the measurements. In this connexion it is instructive to
consider a slightly generalized Michelson experiment in which the whole
apparatus is filled with a refractive medium. From the principle of rela-
tivity it is clear that also in this case no displacement of the inter-
ference fringes will occur when the apparatus is rotated (cf. Fig. 6,
p. 27). This result is also obtained when we calculate the times t[ and t' z
which the two rays take to traverse the paths PS^ P and P*Sf 2 P in the
system of inertia 8' in which the apparatus is at rest ; for in this system the
velocity of light is the same, viz. c/n in all directions and therefore t[ = 2' 2 .
Now it must be required, of course, that the same zero result is ob-
tained when the effect is treated from the point of view of an observer
in a system 8, relative to which the whole apparatus has a velocity v.
The corresponding time-intervals t t and 2 , measured by clocks in S,
may now be obtained from the formula (86) and it is easily seen that
again ^ = 2 . The time which the ray 2 needs to travel from P to
So is obviously \t 2 . During this time the apparatus has moved over
t G. Sagnac, C.R. 157, 708, 1410 (1913), Joutn. de Phys. (5), 4, 177 (1914).
IT, 25
RKLATIVISTIC KINEMATICS
65
a distance \vt z in S, and the ray has traversed a length %ut 2 , where the
ray velocity u is determined by (86) or (47). In Fig. 10 these distances
are shown as PP* and PA!?*, P* and xS* denoting the positions occupied
by P and S 2 after the time i/ 2 . From the figure we see that the angle &
between the ray and the direction of v is given by
cos$ = v/u.
Substituting this in (47) we obtain, by solving this equation with respect
to n 2 , , _. 9V
(95) I $2
By means of the Pythagorean theorem applied
to the triangle PP*8* we obtain, moreover,
from Fig. 10 ,
J? 2 = 15 (96)
(u- V-)*
where / is the distance between the plate P
and the mirror 8 2 . From (96) and (95) we then
get for the time t 2 which the ray 2 needs to
travel from P to 8 2 and back to P
2L
/
>
FIG. 10.
Proceeding to the calculation of the time
which the ray 1 needs to travel from P to S 1
and back to P, we shall here use for the velocity
u the expressions (89), since the ray 1 all the time moves parallel to the
direction of motion of the apparatus. If we denote the time which this
ray needs to travel from P toS l by tf, the mirror $ x will have moved a
distance vt during this time and the total distance which the ray has
travelled relative to 8 is equal to l-\-vtf, where
I = l Q (l~v 2 /c 2 )* (98)
is the distance between P and 8 l measured in S. Therefore we have
l+v/cn l
or
v/cn)
(99)
The time t{ which the ray needs to travel from S l back to S is obtained
simply^ by replacing v by ?;, i.e.
l Q n(l v/cn)
3595.60
66 RELATIVISTIC KINEMATICS II, 25
The total time t l is then
'--'f^-i^v- < 100 >
Thus t l is equal to t 2 , as was to be required.
Even if this result only shows that the theory is consistent, it also
allows us to make a choice between Fresnel's formula (I. 47) and the
relativistic equation (86) which differ from each other in the terms of
second order. If we use (I. 47) instead of (86) in the calculations just
performed, we find for the times t l and t 2 the expressions
which shows that for n = 1 we would obtain t 2 y- t even if we assume
that the distance PS is contracted according to the Lorentz contraction
formula (98).
In the same way as the negative result of the usual Michelson experi-
ment can be regarded as an experimental verification of Lorentz 's
formula (98), a negative result of the corresponding experiment, where
the apparatus is filled with a strongly refractive medium, would mean
a verification of the relativistic formula (86) also as regards terms of
higher order. The same considerations can be applied to Hoek's experi-
ment discussed in 8.
Summarizing, it can be stated that relativistic kinematics in all details
is in agreement with the experimental results. It allows a simple explana-
tion of all dragging phenomena without any ad hoc hypotheses, and it
gives a formula for the Doppler effect which, in contradistinction to the
formula of the ether theory, is in accordance with the experimental
results.
Ill
RELATIVISTIC MECHANICS
26. Momentum and mass of a particle
As mentioned in the conclusion of 18, it is necessary to change Newton 's
fundamental equations of mechanics in order to bring them into accord-
ance with the principle of relativity . In the domain, where all velocities
are small compared with c, relativistic mechanics will, however, go over
into Newtonian mechanics. It is therefore natural to assume that such
fundamental concepts of Newtonian mechanics as momentum and mass
of a material particle also have a meaning in relativistic mechanics.
Therefore, to a material particle moving with the velocity u relative
to a system of inertia S, we shall assign a momentum vector p propor-
tional to u p = mu. (1)
The proportionality factor m is called the mass of the particle. To make
room for the above-mentioned generalization of mechanics we shall not,
however, assume beforehand that m is a constant, but we make the
assumption that m is a universal function, f(u), of the magnitude u = |u |
of the velocity vector, thus
m = m(u)=f(u). (2)
If the velocity of the particle relative to another system of inertia S'
is u', the momentum and mass of the particle relative to S' must be
g iven ^ p'= m'u', (3)
where m' = m'(u') = f(u') (4)
is the same function of to' as m is of u. This follows from the principle of
relativity according to which all systems of inertia have to be treated
on the same footing, so that any relation between physical quantities shall
be form-invariant.
It will now be our primary task to determine the function /. As we
shall see, this function is uniquely determined when we require that the
theorem of conservation of momentum shall hold in any system of
inertia. | Let S and S' be two systems of inertia with the relative velocity
v, and consider a collision between two identical particles 1 and 2 which
before the collision have the velocities u x and U 2 relative to S. The corre-
sponding velocities relative to S' are then determined by (II. 55).
t G. N. Lewis and R. C. Tolman, Phil. Mag. 18, -510 (1909).
68 RKLATIVISTIC MECHANICS III, 26.
Let us now choose the velocities before the collision so that
u; - -u r (6)
By means of (II. 55) it follows then at once that also
u; --= -u 2 . (6)
After the collision the particles will have other velocities, u l and U 2
relative to 8, and u[ and u' 2 relative to 8'. We shall in particular consider
a collision in which the final and initial velocities of particle 1 have
opposite directions, i.e.
Ui = -au t , (7)
where is a positive number. For symmetry reasons we must then also
have
u 2 -= --u 2 (8)
with the same proportionality factor a as in (7), for, according to (5),
the motion of particle 2 relative to an observer in 8' must be the same as
the motion of particle 1 relative to an observer in 8. From (5), (7), and
(8)^ 0' 8 =--fli, (9)
which, according to (II. 55), involves
o; =. -u 2 .
Assuming that the functional relation (2) is the same before and
after the collision, conservation of momentum in 8 gives
/(%)i+/K)2 =/(*i)Ui+/(w 2 )u 2 . (10)
Let us, furthermore, assume that the velocity of particle 1 before the
collision is perpendicular to v, i.e.
(i^.vJ-O. (11)
According to (5), (7), and (9) this involves
K v) -- (12)
and (u t .v) --- (u^.v) -= 0. (13)
For the transformation of the velocities u.> and U 2 we can therefore use
the simple formula (II. 57). Thus we get
u 2 =r u 2 (l~r 2 /c 2 )H v - v-u^l-^/c 2 ) 1 (14)
and 7*5 -= (U 2 .u 2 ) -- ?/f(l ? 2 /c 2 ) + v 2 , (15)
the cross-terms being zero on account of (11).
In the same way we obtain by means of (9)
u.= ul-?; 2
Ill, 26 RELATIVISTIC MECHANICS 69
Introducing (14), (15), and (16) into (10), we get an equation which can
be written
(17)
Multiplying this vector equation by v yields, on account of (1 1) and (13),
Hence, assuming that / is a monotonic function of the argument,
u v u l9 (18)
i.e. a = 1 in (7). This means, on the other hand, that the terms pro-
portional to v in (17) cancel each other and that the coefficients of i^
and u l are equal. Consequently, equation (17) reduces to
[/(i)-/U/[f( -' 2 /f 2 )+' l2 l}(l- 2 /c*) J J(u 1 -u 1 ) - 0. (19)
Since, moreover, according to (7) and (18),
u t D! 2u x -y. o,
the coefficient of u x u t in (19) must be zero, i.e.
/(u,) - (i- v *!c*WU\ul(l-v*lc*) + v*]} (20)
If \ve are to have conservation of momentum in any collisions of the
kind considered, the function / must satisfy the equation (20) for all
values of the independent variables u and v. The solution of this func-
tional equation is obtained by letting u in (20) tend to zero. In this
way we get
It is easily seen that the function f(u) given by (21) satisfies (20) for all
values of u^ and v.
From (2) and (21) we thus obtain for the relativ istic mass of a particle
with the velocity u
m = ___ , (22)
V(i~^ 2 ) ( }
where we have put /(()) m Q .
The constant m Q , the so-called proper mass or rest mass of the particle,
is identical with the mass assigned to the particle in Newtonian
mechanics, and the assumption preceding (10) obviously implies that
the rest mass is unchanged in the collision considered. For the momen-
tum of the particle we now get, according to (1) and (22),
_ m o u
70 RELATIVISTIC MECHANICS III, 27
27. Force, work, kinetic energy
When the velocity of the particje and therefore its momentum are
constant in time, this is taken as an indication that the particle is free.
If, however, the momentum of the particle changes, the particle is said
to be acted upon by a force F which is equal to the change of momentum
per unit time: ,/ n
T? MT I fc"4 A \
F = -. (24)
Equation (24), which for small particle velocities is identical with
Newton's second law of mechanics, should here be regarded as the
definition of a force in the relativistic region. It may be considered as
an equation of motion only when it has been stated how F depends on
the physical state of the system which is the cause of the change of
momentum of the particle.
As in Newtonian mechanics, the work A done by the force per unit
time is defined by A ;= (F.u), (25)
where u is the velocity of the particle. Further, the kinetic energy T of
the particle is defined by the equation
rf j-- A ---(F.u), (26)
at
expressing that the change of the kinetic energy per unit time is equal to
the work A.
Using (23) and (24) the right-hand side of equation (26) may be written
d m n u \ ra n / du\ , m n u 2 du
u . -
or, since ( u . - 7 J = u ,
dt) ' c*(l-~u*/c 2 )i dt
du
u*"\
dt)
A = *^ = [ ^L 1 (27)
/I ,,.2/.,2U J* ,7/1/1 ~.2ls,'2\\r \ I
Introducing this expression into the right-hand side of (26), this
equation can at once be integrated and we obtain for the kinetic energy
of a particle with the velocity u
< 28 >
where C is a constant of integration. Since the kinetic energy may be
taken as zero for u 0, we get
C ~ m c 2 ,
and thus y --/7T m !!/^- m o^ (29)
Ill, 27 RELATIV1STIC MECHANICS 71
If u is small compared with c, we can make an expansion in terms of
(u/c) 2 and we then obtain, to a first approximation, the Newtonian
expression for the kinetic energy
T = %m Q u 2 . (30)
In the same way, all quantities of relativistic mechanics and the relations
between these quantities are identical with the corresponding quantities
and relations of Newtonian mechanics in the domain where the velocities
are small compared with c. It is interesting to note that all deviations
between relativistic and Newtonian mechanics are at least of second
order in u/c, which explains why the early electron theory which was
based on Newtonian mechanics was able to explain all effects of first
order. When the velocity u approaches c, however, the deviations
between relativistic mechanics arid Newtonian mechanics are very large.
For u -> c both the mass (22) and the kinetic energy (29) become infinite,
showing that in mechanics also c plays the role of a limiting velocity.
28. Transformation equations for momentum energy and force
Consider again the two systems of inertia S and 8' corresponding to the
special Lorentz transformation (II. 24). The momentum and the
kinetic energy of a particle are then given in both systems by formulae
of the form (23) and (29). From the transformation equations (II. 45)
for particle velocities we then also obtain the transformation equations
for momentum and energy of a particle.
Now it is convenient to introduce a quantity E defined by the equation
K =-= T+ c* = -,-,-^^T = me*. (31 )
(1U*/C*)*
The quantity E, differing from T by the constant amount m c 2 , is a
measure of the kinetic energy of the particle and frequently we speak
simply of the energy E of a free particle without, however, at the moment
attributing any physical meaning to the constant m c 2 . A similar
quantity E f = T f -\-m c 2 is introduced in the system $'.
By means of (23), (31), and the relations inverse to (II. 45) and to
(II. 48) we then get
u * m Q (l+vu x /c z ) I u\+v
= ^
x <J(l-u 2 /c 2 ) --uc v
m Q (u' r +v)
"-
72 RELATIVISTIC MECHANICS III, 28
In the same way we obtain
A comparison of the equations (32) with the Lorentz transformations
(II. 24') shows that the four quantities
Ps, Pv P S , E\# (33)
are transformed in the same way as the space- time coordinates x, y, z, t.
Thus we have, in analogy to the equations (15), (13), and (13') in Chap-
ter II, /'2
*-%*=? * < 34 )
From (23) and (31) we see that the invariant (34) has the constant value
raj)C 2 . Therefore we have in any system of inertia
E 2
p*--= -m\c\
c
i.e. E = c(ragc 2 +j9 2 )i. (35)
Hence, for the velocity of the particle
Since the quantities (33) also transform like the coordinates (x,y,z 9 t)
by rotations of the Cartesian axes, the transformation equations for
momentum and energy may, in the case of the more general Lorentz
transformation (II. 25'), be written in the vector lorm
p-p ^ !_,-,<.-,
v>Lv ^ A/tl ' 7/c J ! (37)
E ^ Ir7'_l_ / v M
It is immediately seen that (34), which is analogous to (II. 30), is a con-
sequence of (37).
Let us now consider a systemef n free particles. If the momentum and
energy of the ith particle are denoted by p ( *> and E (l) = T (t) -fw ( l) c 2 ,
respectively, where T^ is the kinetic energy and w ( is the rest mass of
the ith particle, the total momentum and energy of the system are de-
fined by n
^ 2P< "' - - : <,
rr\ V TO) m - V
1 2, J- > ni () 2,
i i
Ill, 28 KKLATIVISTIC MECHANICS 73
Since the transformation equations (32) or (37) are valid for^each
particle separately and as they are linear transformations, it is obvious
that the same equations are valid also for the total momentum and
energy of the system. Thus we can take over the equations (32) and (37)
for a system of free particles, where p, E, and T denote the total momen-
tum and energy, while w? , according to (38), is the sum of the proper
masses of the particles. From the transformation equations (32) and
(37) it is immediately seen that, if the theorem of conservation of
momentum in a collision between the particles is valid in every system
of inertia, the total energy E must also be conserved in any system of
inertia.
Returning to a system consisting of a single particle, the equations (24)
and (26) may be written in the form
d j t = F, (39a)
dt
^ = (F.u). (396)
Since the equations (39) are to hold in any system of inertia we can
deduce the transformation equations for the force F from the known
transformation properties of the quantities on the left-hand side of (39)
From the equations (II. 25') and (II. 26) we get
dt _ H (v.u')/r 2
Hi' ~ ~J(i^*lc*) '
Thus, by means of (37),
F dp dt'
^ dt' dt
')[l^^
~~ - '
where F' is the force in the system S'. In relativistic mechanics the con-
cept of force has no longer any absolute meaning as it has in Newtonian
mechanics.
If we introduce the proper time T of the particle instead of t and the
Minkowski force ^
> \ I/
the equations (39) can, by means of (II. 38), be written
dp _, dE
p M , :il = f . u. (42)
dr dr
74 RKLATIVISTIC MECHANICS III, 28
Since r is an invariant, the quantities {p,E/c 2 } on the left-hand sides
of (42) transform like the space-time coordinates {x,}: the same must
be true, therefore, for the quantities {F/, /5 (F u .u)/c 2 }. This can also be
shown directly by an elementary calculation from (40), (II. 55), and
(II. 56).
29. Hyperbolic motion. Motion of an electrically charged par-
ticle in a constant magnetic field
As mentioned before, the equation (39 a) can be regarded as equation
of motion only when it is known how the force F depends on the variables
of the physical system causing the change of momentum of the particle.
If the velocity of the particle is small relative to c, the relativistic equa-
tion must, however, be identical with Newton's second law and, in the
inertial system 8 relative to which the particle has velocity zero at the
moment considered, we may assume that the force F is identical with
the Newtonian force. By means of the transformation equations (40)
we can then calculate the force F in an arbitrary inertial system S. Let
the velocity of the particle relative to S be u, if S' in (40) is the rest
system $, we have v u and u' = and we then get for the force
F in S (u
F -* a l <
where F is the Newtonian force. Decomposing the forces into two com-
ponents respectively, parallel and perpendicular to u, thus putting
F = F,i4-F, F F?4-F^
* * III* J_ A -~ J. || -fJ. j_,
(43) can obviously be replaced by the simple equations
F,,=--F{|, F L = F^l-* 2 /' 2 ) 1 - (43')
If we know the Newtonian force F we can thus, by means of (43) or
(43'), calculate F in any system S. In this way we can show, for example
(Chap. V, 58), that the force acting on an electrically charged particle
travelling with the velocity u through an electric field E and a magnetic
field H is given by the Lorentz formula
F- e E + -(uxH) , (44)
where e is the electric charge of the particle and u X H denotes the vector
product of the vectors u and H.
Now the left-hand side of the equation of motion (39 a) can, by means
of (1), (31), and (396), be written
dp d(mu) du dm du 1 dE du ,
* ' '_ tyy\ I II <yy\ I || tyY* I
dt ~ dt ~ dt dt ~~ dt c 2 dt ~ dt
Ill, 29 RELATIVISTIC MECHANICS 75
where m is the relativistic mass given by (22). Introducing this into
(39 a) we get the equations of motion in the form
m du F () (45)
From this we see that the acceleration of the particle in general has a
direction different from the direction of the force and, therefore, the
motion of the particle is more complicated in relativistic mechanics than
it is in Newtonian mechanics. Only if the force is constantly parallel
or constantly perpendicular to u, will the motion of the particle again
be relatively simple. We shall here treat an example of each of these
cases.
Let us first consider a particle which is acted upon by a constant force
F = m Q g and which has an initial velocity in the direction of the force.
According to (45) the particle will then continue to move in the direction
of the force. Therefore the path of the particle will be a straight line and
we can choose this line as x-axis. (39 a) then reduces to
u \ F dx
If we assume that the velocity is zero at the time / -- 0, \\e get by a
first integration
dx at , An .
B = = - (46)
If we further assume that x when t ~ 0, we obtain by a second
integration
<47)
or i ! | ~ v .
ff 2
If we plot this motion in an ^/-diagram the equation (48) will represent
a hyperbola and, consequently, this motion is called a hyperbolic
motion. |
As long as (gt) 2 < c 2 , we can neglect powers of (gt/c) higher than the
second, and we obtain from (47) the usual equation
x :== - T?gt
representing the motion of a particle with constant acceleration.
t M. Born, Ann. d. Phys. 30, 1 (1909); A. Sommorfeld, ibid. 33, 670 (1910).
76 RELATIVISTIC MECHANICS III, 29
For large t, i.e. for large velocities, on the other hand, the increase in x
with increasing t is slower than that according to Newtonian mechanics.
For t > oo the velocity u given by (46) approaches the finite value c
independently of the value of g\ thus, even if a particle is attacked by a
very large constant force, it will never attain a velocity exceeding the
velocity of light. This is in agreement with the considerations in 21,
p. 53. An electrically charged particle moving in a constant electric
field with a velocity parallel to the direction of the field represents a case
of the type just considered.
Let us now consider the motion of a charged particle in a constant
magnetic fidd H. Decomposing the velocity u of the particle into com-
ponents parallel and perpendicular to H respectively, thus putting
u ~ U||-f-Uj_, the force acting on the particle may, according to (44), be
written
F = ^(uxH) = (u xH), (49)
c c
i.e. the force is perpendicular both to u, u L , and H. Therefore,
(F.u) = 0,
and (45) assumes the same form as in Newtonian mechanics, i.e.
m-" = F = P (Uj.xH). (50)
at c
Here in denotes the relativistic mass which, however, in our case is
constant, since K and thus also m and u, according to (39 6), are constant
in time.
From (50) it follows that the component of the acceleration in the
direction of H is zero, i.e. the component of the velocity in the direction
of the field, viz. u\ { \, is constant in time and, since also u = |u | is constant,
the same must be the case for the magnitude IUJL | of the vector Uj .
Therefore the path of the particle must be a helix having its axis in the
direction of the field The projection of the path on a plane perpendicular
to H will be a circle with radius p determined by the condition that the
centripetal force mu\jp in the circular motion must be equal to the force
(49). Therefore, we have
mu 2 \ e Tr
- =L -u\H
or p L = mu< = -Hp. (51)
Ill, 29 RELATIVISTIC MECHANICS 77
If the velocity of the particle is perpendicular to the direction of the field
we get simply
p = -Hp. (52)
c/
This equation enables us to determine the momentum of a charged
particle by measuring H and p. This method has been especially useful
in cosmic ray analysis and in j3-ray and mass spectroscopy.
30. Equivalence of energy and mass
Let us again consider the system 2 X of n free particles. If (p, E) and
(p',E') denote the total momentum and energy of the system in the two
systems of inertia S and 8', respectively, the connexion between the
primed and unprimed variables is then given by (32) or (37). The in-
variant (34) will for such a system always have a negative value. For
n =. I this invariant is, according to (35), equal to w 2 c 2 and, keeping
in mind that (35) holds for each particle, it is immediately seen that
p 2 E 2 /c 2 must be less than -]T ( m o >c ) 2 f r n > * This means that we
i
can always choose an inertial system 8' such that the total momentum
p' of 2 t is zero in 8'. Viz., putting p' in (37), we obtain for the
relative velocity v of 8' and 8
v-^P (53)
and, since p 2 E 2 /c 2 < 0, we have E > cp so that the relative velocity v
determined by (53) is less than c, which must be required.
Such a system of inertia 8 in which the total momentum p is
called the rest system of X t or the centre of gravity system since the
system Xj as a whole has the same mechanical properties as a particle
at rest relative to 8. Now let u be the velocity of 8 relative to an
arbitrary system of inertia 8, then u also denotes the velocity with which
the system i^ as a whole moves relative to S.
If the s\ stem 8' in (37) is identical with the rest system 8, we get the
equations ^ ^ ^ ^
P - r \/(T-i2/f~V ~~ J(T-u*lc*)' ( }
which express the total momentum and energy of i^ as a function of the
velocity u of the system. It now seems natural to define the total mass
of the system as the ratio between momentum and velocity in the same
way as for a single particle (cf. equation ( 1 )). From (54) we then see that
we must assign to i] t a mass M given by
M = ^1* = - -JL (55)
** *~ ** v '
78 RELATIVISTIC MECHANICS III, 30
This corresponds to a rest mass M Q = E/c 2 which, by means of (38),
can be written in the form
M Q - EO/C* = m +T/c 2 , (56)
where m is the sum of the rest masses of all particles, and T denotes the
total kinetic energy of the particles in the rest system.
With the expression (56) for the rest mass of Sj the equations (54) for
the momentum and energy of 2 X become completely analogous to the
equations (23) and (31) holding for a single particle. From (56) we see that
the rest mass of S^ is larger than the sum ra of the rest masses of all
particles by the amount T/c 2 . Thus the inner kinetic energy of the
system contributes to the inertial mass of the system by an amount
equal to T/c 2 .
The invariant in (34) can, by means of (54) and (56), be written
p2-E*/c* = -Mlc* (57)
in analogy to the equation (35) valid for a single particle. Equation
(57) may also be used as a definition of the rest mass of D 1 .
The important conclusion that the inner kinetic energy of a system of
free particles corresponds to an inertial mass can now be extended to any
kind of energy.
Starting from the assumed general validity of the conservation
theorems,
&m = ~. (58)
c
To prove this important theorem consider a 'collision' of the system
E! of free particles with another arbitrary physical system S 2 by which a
certain amount of energy and momentum is transferred from 2 X to S 2 ..
Before and after this collision process the particles in Sj are free; there-
fore the total momentum and energy of S t before and after the process
are transformed according to equations of the form (32) and (37).
Subtracting now the transformation equations for the energy and
momentum after the 'collision' from the corresponding transformation
equations before the collision, we get if (Ap, AA 7 ) and (Ap', AJ?') denote
the difference between the momentum and energy of EJ before and after
the process measured in S and S', respectively,
(59)
A =
Ill, 30 RELATIVISTIC MECHANICS 79
If momentum and energy are to be conserved during the process the
momentum and energy of the system 2 2 must be increased by the
amounts Ap and A7 in S and by Ap' and AJ5' in S'.
In analogy to (34) it now follows from (59) that the quantity
is an invariant. When this invariant has a negative value, it is again
possible to find a system of inertia S' = 8 in which Ap' == Ap 0,
and if u is the velocity of S relative to S, we get from (59)
u A//r 2 A K
A comparison of these equations with (-2) and (23) shows that we must
assign to the energy &E transferred to H 2 a rest mass
(02)
C"
and a mass Am relative to 8, which is
for the change of momentum of 2 2 during the process is the same as if a
material particle with the rest mass Am and the velocity u relative to
S had been added to the system X 2 . The invariant (60) can now be written
|Ap|2_^|l 2 ^ _(Am ) 2 c 2 (64)
in analogy to (35). This equation gives a simple expression for the rest
mass of the transferred energy.
Since H 2 was a general physical system so that the transferred energy
may have any form, we see that formula (63) must hold for any type of
energy. The system 2 2 may, for instance, be an electromagnetic field such
that the energy transferred from Sj has taken the form of electromagnetic
radiation. Thus the transformation equations for the energy and momen-
tum of electromagnetic radiation must be given by (59). Further, if
Do is a body which transforms the absorbed energy AA 1 into heat, we see
also that the heat energy of a body contributes to its mass, so that the
mass of a body is increased by heating. If, finally, 2 2 were a system
converting the transferred energy into potential energy, it follows that
inertial mass has to be assigned also to the potential energy of a system.
From the preceding argument, in particular from equation (64), it
follows that the notion of mass of a certain amount of energy E has a
80 RELATIVISTIC MECHANICS III, 30
well-defined meaning only if we also know the momentum Ap connected
with the energy, and for this mass to be real the left-hand side of (64)
must be negative. Only for
|A#| >c|Ap| (65)
may we speak at all of a rest system and, consequently, of a certain
velocity and of a real mass of the energy.
In the discussion above E may be negative, so that we actually have
to deal with a transfer of energy to the system S^ Thus, considering a
process in which the total momentum p 2 and energy E 2 of 2 2 are trans-
ferred to X 1 , we see that the transformation equations (59) must hold for
the total momentum and energy of an arbitrary system. If, furthermore,
the relation
p*-~< (}
holds, the system has a real rest mass defined by (64), i.e. by
p|_. (67)
C"
In the case where > p\'K\l< t ^ is equal to zero the rest mass of the system
will also be zero.
If H 2 is a system of fields where the density of field energy is a homo-
geneous positive definite function of the field variables, (66) must always
be fulfilled. If E 2 were less than cp 2 , we could find a system of inertia 8'
in which E' 2 =- and p 2 ~/~ 0. For this purpose it would be sufficient to
give v in the transformation equations (59) for p 2 and E 2 the value
v-A' 2 P 8 /pI (|v|<c). (08)
According to the relations inverse to (59) we should then have
According to the assumptions made above regarding the dependence
of the energy density on the field variables, E' z can, however, be zero only
if the field itself vanishes and in this case the momentum of the field
p' 2 must also be zero. It is therefore natural to assume that the relation
(66) holds for any macroscopic physical system. As a simple example
of a system where the sign of equality holds in (66) we have the case of a
train of electromagnetic plane waves. According to (67), such a system
must therefore have the rest mass zero and a velocity u c 2 p 2 /E 2 = c
in any system of inertia.
Now it may also be shown that, conversely, any material particle with
Ill, 30 UKLATIVISTIC MECHANICS 81
the mass m must represent an energy E = me 2 and in particular in the
rest system of the particle an energy E = m c 2 . This statement ob-
viously has a real meaning only when a process exists in which the
energy represented by the mass of the particle can be transformed into
another form of energy such as the kinetic energy of other particles.
We cannot know in advance whether such an 'annihilation process'
actually does exist in nature, but we can show that if it exists under
certain conditions and if we require that the principle of relativity as
well as the theorems of conservation of momentum and energy shall
hold for this process, the amount of energy liberated by annihilation of
the mass m will be equal to E = m c 2 .
To prove this statement, we assume that the momentum and energy
liberated by the annihilation process are transferred to the previously
considered system H 1 of free particles. Consider again two systems of
inertia 8 and /S, where 8 is the rest system of the particle. If (Ap, Ai7)
and (Ap, A/) denote the transferred momentum and energy measured
in S and $, respectively, we have, according to (59)
Ap = Apo-, j
U~ ^J(1U 2 /C")
where u is the velocity of the particle relative to 8.
Since the momentum of the particle is zero in the rest system and
since, moreover, momentum and energy of the particle are zero in any
system of inertia after its annihilation, we obviously have
Ap = 0, A# = E , (70)
where E Q denotes the unknown energy content of the particle at rest
before the annihilation. Furthermore, Ap is equal to the momentum p
of the particle relative to 8 before the annihilation :
A m n u
Substituting from (70) and (71) in (69) we obtain
ra u __ E u/c 2
or E Q = m c 2 . (72)
The total energy content of a free particle with the velocity u must
then be 2
E = E Q +T = m C 2 = me 2 , (73)
3595.60 n
82 RELATIVISTIC MECHANICS III, 30
where we have made use of (29) and (22). The quantity E introduced
in (31) has thus a deeper physical meaning. The energy (72) is called
the proper energy or rest energy of the particle.
We have thus obtained a general proof of Einstein's famous formulaf
E = me 2 , (74)
which states that any energy E has an inertia corresponding to a
mass E/c 2 , and that any mass m represents an energy me 2 . This theorem
of the equivalence of energy and mass is one of the most important
results of the special theory of relativity. It should be noted that the
mass involved in this theorem is the inertial mass. One of the basic
assumptions of the general theory of relativity is, however, (cf. Chap.
VIII, 83), the equality of inertial and gravitational mass, so that we
shall have to ascribe to an energy of amount E also a gravitational mass
m given by (74).
31. Inelastic collisions. Mass of a closed system of particles
Before proceeding to a discussion of the experimental verification of
relativistic mechanics we shall, in the present section, consider a few
simple examples illustrating the general theorem of the equivalence of
mass and energy
Let us first consider a completely inelastic head-on collision between
two clay balls of the same rest mass m Q which in the inertial system
S' $ travel along the same line with velocities of equal magnitude but
opposite in direction. The total momentum and energy in the centre of
gravity system # before the collision are then
p o ^ 0} //;o ^ 2m c 2 +T, (75)
where T Q is the total kinetic energy. In a system of inertia S relative to
which iS' has the velocity u, we thus have, according to (37) and (75),
for the total momentum and energy before the collision
(2m +2*>/c 2 )u
By the inelastic collision, the two clay balls will unite and form one large
ball ; on account of the theorem of conservation of momentum applied
in /S, this ball will have zero momentum and thus also zero velocity in 8.
t A Kmstoin, Ann d Phys. 18, 639 (1905), 23, 317 (1907), H. A. Lorentz, Das
JRelativitatspnnzip 3 Haarlemer Vortrage, Leipzig 1914, id, Amal. Versl. 20, 87
(1911).
Ill, 31 RKLATIVJLST1C MECHANICS 83
The original kinetic energy T of the particles in S is converted into heat
and, on account of the conservation of energy, the amount of heat
developed is Q Q - 710 /77\
The day ball after the collision now has the velocity u relative to S,
i.e. a momentum M _.
where M is the rest mass of the ball after the collision. As the conserva-
tion of momentum theorem must hold also in S, the expressions (78)
and (76 a) must be equal. Thus, applying (77), we get
Jlf -----L> -| Q , (79)
\j
i e. the rest mass of the ball M {} after the collision is equal to the sum of
the rest masses of the original balls increased by the mass of the heat
energy.
For the energy of the ball after the collision in tf, we obtain, according
to (79) and (77),
F - M C * - ~" ? <> C ' M T
~'
which is equal to the total energy (7()/>) of the balls before the collision,
in accordance with the theorem of conservation of energy in #. For the
difference between the total kinetic energy before and after the collision
we obtain, however,
I i\ no
^ =
(81)
by means of (7(5/>), (79), and (77). This difference is thus an invariant
independent of the system of inertia in which the kinetic energies are
calculated.
To give an illustration of the inertia of potential energy we now con-
sider a system X consisting of a certain number of particles which are
held together by attractive forces. Let us assume that there exist
inertial systems #' in which all particle velocities are small compared
with the velocity of light such that in *V \\e can use the non-relativistic
Newtonian mechanics as a good approximation. Neglecting the typically
atomic phenomena connected with the existence of Planck's quantum of
action, we may consider an atomic nucleus to form such a mechanical
system, the atomic nuclei being built up of elementary particles, the
nucleons, which are so heavy that their velocities can generally be
84 RELATIVISTIC MECHANICS III, 31
regarded as small compared with c. This assumption means that the
proper times of the separate particles in 2 are practically identical and
equal to the time /' in S' and, furthermore, that the forces between the
particles can be considered to a first approximation to be functions of
the positions of the particles. If, moreover, the forces are conservative
forces, they can in S' be expressed as gradients of a potential function V
which is a function of the position coordinates of the particles. According
to Newtonian mechanics the particles in the system will now move in
such a way that the sum of the kinetic energies and the potential energy
is constant in time, i.e.
T' + V =-- //' - constant. (82)
However, the total kinetic energy is, of course, not constant in general.
We now adjust the arbitrary constant in the potential energy V so
that V -- when the particles are so far from each other that the forces
are zero. Thus V is obviously ^negative in any state where the particles
are bound to each other.
Among the possible systems of inertia S' we now choose that system
S in which the centre of gravity of X is at rest. The sum p of the
momenta of the particles is then obviously constant and equal to zero
in H. In a system of inertia 8 relative to which /S is moving with the
velocity u in the direction of the #-axis, the sum p of the momenta of the
particles is, according to (32), given by
where again w? is the sum of the rest masses of all the particles. Since the
proper times of the particles of the system are practically identical, p
can be regarded as a function of a well-defined time variable. In contrast
to the case of a system of free particles, p is not constant in time since
the kinetic energy T occurring in the expression (83) for p is time-
dependent.
However, p is not the total momentum of S in S, since we have to take
into account the fact that the potential energy V has a rest mass V/c 2 .
The total momentum P in S is therefore
where 7/ T-\- V is the total energy in the rest system. In contrast
to p, the total momentum vector P is obviously constant in time,
as is required for a closed system with no external forces. If the
Ill, 31 RELATIVISTIC MECHANICS 85
particles are so far away from each other that they can be regarded as
free, we must have P - p, a condition which determines uniquely the
constant in the potential energy.
From (84) we see that an atomic nucleus which moves as a whole with
the velocity u relative to a system of inertia 8 has a total momentum
" = ;,->
with a rest mass M Q ^ m Q +H/c 2 . (86)
For stable nuclei // is negative and A A 1 ^- 7/ is the binding energy
of the nucleus, i.e. the amount of energy which must be transferred to
the nucleus in order to disintegrate it completely into its constituent
particles. For the quantity Am = W Q 3/ , the mass defect of the nucleus,
we ha\e, according to (86),
Am = A ^. (87)
c 2
This fundamental relation connecting the binding energy and the mass
defect, which is a special case of Einstein's formula (74), has now been
verified experimentally with great accuracy in many nuclear reactions
(cf. 32).t
32. Experimental verification of relativistic mechanics
In 26 we saw that, if the theorem of conservation of momentum is to
hold in any elastic collision between two particles, the variation of the
mass with velocity must be given by (22). The next question is whether
something like conservation of momentum and energy exists at all for
large velocities. This question can only be settled by experiments.
Direct experiments of this kind were performed by Champion, J who
investigated the collisions between rapidly moving electrons (j3-particles)
and elecl rons at rest in a Wilson chamber. Let us consider more closely
such a collision between an electron 1 with velocity U 1 and momentum
p l = raQU^l ftf/c 2 )-* and an electron 2 with velocity zero, i.e. with
the momentum p 2 = ) elative to a systom of coordinates S in which the
cloud chamber is at rest. In the collision a certain momentum is trans-
ferred to particle 2. Let p l and p 2 be the momenta of the two particles
after the collision, and 6 and <f> the angles between the direction of the
incident electron and the vectors p l and p 2 , respectively (cf. Fig. 11).
t The possibility of such an effect was first discussed by P. Langevin, J. de Phys. (5),
3, 553 (1913).
J F. C. Champion, Proc. Roy. Soc. A, 136, 630 (1932).
86
RKLATIVTSTIC MECHANICS
III, 32
Application of the conservation laws of relativistic mechanics then leads
to a simple relation between 6 and <, which we shall now deduce.
It is convenient here to make use of the fact that the theorems of
conservation of momentum and energy are valid in any system of inertia
(if they hold at all). For convenience we choose the Cartesian axes in
U
KKJ. 11
the laboratory system 8 such that p t is parallel to the #-axis and p t lies
in the #y-plane. Then, if the momentum is conserved, p 2 also must lie
in the a:/y-plane. We now introduce the centre of gravity system S' in
which the total momentum p' pi+Pa -= 0. 8' is moving relative to 8
with a certain velocity v in the direction of the .r-axis. From the reciprocal
equation to (32 a)
which holds for each particle separately as well as for the total momen-
tum p and energy E of the system, we then get, since p -- p l and
E = E^m^c 2 m c 2 +m c 2 (l~^f/r 2 )"s the following expression for v:
c Pi
v
E
Since p' 2 -- pi, the two particles have the same initial velocity u'
relative to 8' '. Since, furthermore, particle 2 was at rest in S, the velocity
u' relative to 8' must be equal to the relative velocity r of 8' and 8, i.e.
u v,
IE'
(89)
Applying now the theorems of conservation of momentum and energy
in the centre of gravity system, we obtain for the momenta and energies
Ill, 32 RELATIVISTIC MECHANICS 87
of the particles after the collision
I>2 = Pi E'i = E* = \E' = -/,-, ..g/-^
Pa =-- Pi - ._
Consequently the two particles will also have the same velocity v relative
to S' after the collision and, according to (90), we have
-*?=*. <..,
From Fig. 1 1 we see at once that
tan 9 ~- 4~ , tan <f> = _ 2 " .
Plx P2jr
Using the transformation equation (32) for the momenta of the two
particles after the collision, we obtain by means of (90) and (91)
l ;
Introducing the expression (88) for ?; into (92) we finally get
tan0tan<)& = " , (93)
71+ 1
where we have put -y l = (lul/c 2 )-*. (94)
Thus, the product of tan0 and tan^ is independent of the rest mass of
the particles (as long as they are equal) and is a function of the velocity
of the incident particle alone.
In the limit c, -> oc, we obtain the corresponding formula of Newtonian
mechanics. In this limit,
y l 1 and tan#tan</> = 1,
.e.
or 6+<f> = ITT. (95)
According to Newtonian mechanics the directions of motion of the
two particles after the collision should thus be perpendicular to each
other. This well-known effect can be observed in collisions between
billiard balls, for instance, and has also been verified in collisions between
a-particles and helium nuclei in a Wilson chamber, the velocity of the
88 RELATIVISTIC MECHANICS HI, 32
a-particles being so small compared with c that Newtonian mechanics
may be applied in this case. When the velocity of the incident particle
approaches the velocity of light, however, y l > 1 and, thus,
tan 6 tan cf> < 1,
which shows that the angle between the directions of motion of the
particles after the collision in this case will be smaller than \TT.
This characteristic phenomenon is well suited to an experimental test
of relativistic mechanics. Cloud-chamber pictures of the collision of
j8-particles with electrons at rest in the chamber allow a direct determina-
tion of the angles 9 and <. It turns out that in many cases #+</> is con-
siderably smaller than 90. If, in addition, the velocities, and consequentlv
also y 1? for the incident /?- particles are known, we arrive at a direct test of
the relativistic formula (93). Measurements of that kind were performed
by Champion, *)* who found agreement with formula (93). This may be
regarded as an experimental proof of the theorems of conservation of
momentum and energy in collisions between electrons, and thus also
of the relativistic formula (22) for the variation of mass with velocity.
The accuracy of these measurements was quite sufficient to disprove
the formula for the variation of mass with velocity put forward by
Abraham} before the development of the theory of relativity.
The equations of motion (39^) and (44) can also be tested directly by
measuring the deflexions of rapidly moving electrons in electric and
magnetic fields. In the case of a constant magnetic field perpendicular
to the direction of motion of the electron, the previously derived equa-
tion (52) should hold, i.e.
mU = H ?. (96)
e c
If, on the other hand, the electron traverses a constant electric field
perpendicular to the initial velocity u of the electron, the electron will,
according to (45) and (44), have a constant acceleration g eE/m in
the direction of the field as long as the deflexion from the original path
is small. In the time t = l/u which the electron requires to traverse a
distance I in the field, the electron is thus deflected a distance
-*-*
perpendicular to the initial direction of motion.
If, therefore, the electron traverses both a magnetic and an electric field
of the above-mentioned kind, simultaneous measurements of //, p, E, I,
f See ref., p. 85. % M. Abraham, Ann. d. Plnjs. 10, 105 (1903).
Ill, 32 RELATIVISTIC MECHANICS 89
and AZ allow us to determine the variation of m with u by means of (96)
and (97). Such experiments were performed by a large number of investi-
gatorsf and the results obtained in the later experiments are in close
agreement with the relativistic formula (22). Simultaneously the
experiments may be regarded as a test of Lorentz's expression (44) for
the force acting on an electrically charged moving particle.
If we know the charge of the particle from other experiments, measure-
ments of the deflexion of the particle in known electric and magnetic
fields allow us to determine the absolute value of the mass of the particle.
This is used in the so-called mass spectrograph, which permits a .very
precise determination of the mass of atomic nuclei.
While thus the relativistic equations of motion of fast charged particles
were experimentally proved at an early time, it was not until recent years
possible to test the equations (74) and (87) which express the equivalence
of energy and mass. This is understandable if we bear in mind that the
change in mass of a body due to its potential energy or due to heating
according to these equations in general will be negligible compared with
the total mass of the body. The situation is different, however, when
Einstein's relation is applied to single atomic nuclei.
From the results obtained at the end of the preceding section we should
expect the mass of an atomic nucleus in its ground state to be always
smaller than the sum of the masses of the constituent nucleons. Accord-
ing to (87) this mass defect should be particularly large for the stablest
nuclei with large binding energy. Measurements of nuclear masses by
means of the mass spectrograph have confirmed this result and in some
cases the mass defect amounts to several per cent, of the mass of the
whole nucleus.
As soon as one succeeded in producing nuclear reactions in which
nuclei with a given mass defect are transformed into nuclei with other
mass defects it also became possible to test Einstein's formula (74)
experimentally. Here we shall only discuss one single example of such a
process. When lithium is bombarded by fast protons (Oockcroft and
Walton, 1932J) it happens that a proton (}H) penetrates into, a lithium
nucleus (JLi), thus forming a compound nucleus which is unstable and
quickly splits into two fast ex-particles (|He). This process can be
t Among the numerous paper** wo quote W. Kaufmann, Gott Nadir, niath.-nat.
Klasse, p. 143 (1901); A. H. Bueheioi, Veth. d. Dculxchen Phy*. Ges. 6, 688 (1908),
G. Neumann, Ann. d. Phys. 45, 529 (1914); Ch. K. Guye and Ch. Lavanehy, Arch, de
Geneve, 41, 286, 353, 441 (1916). See also the comprehensive account by W Gcrlach
in Hdb. d. Phys 22, 61 (1926).
J J. D. Cockeroft and G. T. S. Walton, Pioc. Roy. floe. A, 137, 2J9 (1932).
90 RELATIVISTIC MECHANICS III, 32
expressed by the equation
IU+\H - JHe+JHe. (98)
The masses of the nuclei entering in this process are known very well
from mass-spectrograph measurements. According to the latest measure-
ments the mass of Li is 7 1 66 if the mass of the oxygen atom is put equal
to 16. Now, if we also content ourselves with four decimal places for
the hydrogen nucleus and the helium nucleus, the corresponding figures
for JH, |He are 1-0076 and 4-0028, respectively, f Thus the loss in mass
m during the process is
m = 7-0166+1-0076 2x4-0028 0-0186
mass units, or in grammes,
m = 0-309 X 10- 25 gin. (99)
According to Einstein's relation this mass represents an energy of
me* - 27-7 x 1 0- erg, (100)
which after the process must appear as kinetic energy of the a-particles
By means of the range-energy relation the kinetic energy of the particles
can now be determined from measurements of the ranges of the ex-
particles. Newer very precise measurements^ give for the difference
between the total kinetic energy of the a-particles and the kinetic energy
of the incident proton the value of
E !7-28()-03MeV --- (27-60-()5)x 10~ 6 erg. (101)
The agreement between (100) and (101) is excellent. The deviation is
less than the possible error in the accuracy of the measurements, the
error in the determination of the mass of the lithium nucleus involving
an error in me 2 of 0-2 x 10- 6 . Einstein's equation (74) can thus be
regarded as being verified experimentally with high accuracy, the error
being less than 1 per cent.
While the mass corresponding to the heat developed during a usual
chemical process is immeasurably small, the amount of heat developed
in a nuclear reaction pile is so large that the corresponding mass may be
of the order of many grammes. Tri the fission of a uranium nucleus,
where the nucleus is divided into two fragments, an energy A A 7 is
developed whose mass Am = A//c 2 amounts to about a thousandth of
the mass of the uranium nucleus. By complete fission of all the nuclei
in 1 ton of uranium the mass of the heat developed would thus be of the
order of magnitude of 1 kg.
f K. T Bambndge arid E. B. Jordan, Phy.s. Rev. 51 , ,384 (1937) , .see aho H. Bethe and
M. S. Livingston, Rev Mod. Phys 9, 370 (1937).
J N M. Smith, Jr., 7%*. Rev 56, 548 (1939).
Ill, 32 KELATIVISTTC MECHANICS 91
Modern atomic physics, on the other hand, could also verify the
equation (72) according to which a particle of mass w represents an
energy E Q m c 2 . This statement can be tested only (and therefore has
a real meaning only) if a process exists in which the particle is annihilated
completely. After the discovery of the positive electrons, the positons,
in 1932f it became clear that such annihilation processes do exist in
which a positive and a negative electron (a positon and a negaton) are
annihilated, in accordance with Dirac's theory of electrons. J Since both
particles have the same mass m^ the amount of energy liberated in such a
process should, accoiding to (72), be equal to 2w r 2 . This energy is
emitted in the form of electromagnetic radiation and by measurements of
the energy of this radiation equation (72) could be verified. The reverse
process, in which electromagnetic radiation (light quanta) is transformed
into pairs of positive and negative electrons, is also possible provided the
light quanta participating in each individual process have the necessary
energy of 2r/? c 2 .
Summarizing, we can state that all consequences of relativistic mech-
anics have proved to be in complete agreement with the measurements.
The objection which could be raised against this statement is that all
these measurements were performed on atomic particles, electrons and
nuclei, etc., which in view of the existence of the quantum of action
cannot be treated by means of classical mechanics. Since, on the other
hand, classical mechanics is only a special case of quantum mechanics,
the experiments mentioned above simply show that such consequences
as the theorem of the equivalence of energy and mass must be valid
beyond the domain of classical mechanics. In the general proof of this
theorem, given in 30, the system H 2 was in fact a quite general physical
system without any limiting assumptions regarding its constitution.
Moreover, in all experiments on the deflexion of electrons in a macro-
scopic electromagnetic field we are concerned with a case where quantum
mechanics goes over into classical mechanics so that these experiments
can be regarded as a direct verification of the fundamental equations of
classical relativistic point mechanics.
f C. J>. Andeison, faience, 76, 238 (11)32) , /'////*. Her. 43, 4!H ( 1 933) , P. M. S Blockrtt
and G P. S. Oochialmi, I 9 tor Rot/. Noc A, 139, 699 (1933)
J P. A. M. Dirac, The Principle* of Quantum Mechanic*, 3id ed., (Kfoid 1947, 73.
C D. Anderson and S. H. Noddei moyoi , Pht/tt lt<v. 43, 1034 (1933). See also
K. Kasetti, L. Meitner, and K Philipp, Natunv. 21, 286 (1933), J. Cune and F. Johot,
CM. 196, 1581 (1933).
IV
FOUR-DIMENSIONAL FORMULATION OF THE
THEORY OF RELATIVITY: TENSOR CALCULUS
33. Four -dimensional representation of the Lorentz trans-
formation
IN a definite system of inertia S an arbitrary event is characterized by the
four space-time coordinates (x,y,z,t). In another system of inertia S'
the same event is characterized by four other numbers (#',?/', 2', '). If
we assume that the origins of the ( 'artesian coordinates in the two systems
S and 8' coincide at the time t = t' 0, the connexion between these
space-time coordinates is given by a homogeneous Lorentz transforma-
tion, i.e. by a homogeneous linear transformation leaving the quantity
s 2 (II. 13) invariant, i.e.
s 2 = x 2 +y 2 +z 2 c 2 t 2 = x' 2 +y' 2 +z' 2 c 2 t 2 s' 2 . (1)
u in ^ x l x, x 2 y, a? 3 ~ 2, # 4 = ict
x'i = x', x' 2 = y' 9 x' 3 = 2', x\ = ict',
where i = <j( 1) is the imaginary unit, the homogeneous Lorentz trans-
formation can thus be characterized as a homogeneous linear transforma-
tion 4
(=1,2,3,4), (3)
k- 1
which satisfies the condition
=!>;= 2 x?. (4)
i i
Here the coefficients oi ik depend only on the angles between the spatial
axes in S and S' and on the relative velocity of the two systems of inertia.
Since the coordinates x 1% x 2 , x 3 are real, and # 4 is purely imaginary in
every system of coordinates,
oc iK and a 44 are real,
while a l4 and 4/c are purely imaginary,
if i and K denote arbitrary index values between 1 and 3.
If the variables (x t ) and (x\) were real numbers, they could be inter-
preted as Cartesian coordinates of a point in a four-dimensional Euclid-
ean space. Then the transformation (3) would represent a simple rotation
of the Cartesian system of coordinates in this space, the distance
IV, 33 THE THEORY OF RELATIVITY 93
(4) from the observed point (#J to the origin (0, 0, 0, 0) being invariant
by such a rotation.
Now it is natural, even if # 4 and x\ are not real, to introduce a four-
dimensional space whose points are defined by the coordinates (xj.
Since any event in physical space is characterized by a definite set of
numbers (# t ) in every system of space-time coordinates, each event is
thus depicted in a definite point of this abstract four-dimensional space.
This space, which was first introduced by Poincaref and Minkowski,J
is called the space-time continuum or simply the four-space or (3+1)-
space, hereby recalling that the four dimensions of the space are not com-
pletely equivalent. A homogeneous Lorentz transformation (3) can
thus be interpreted as a rotation of the system of coordinates in the
(3+ l)-space. The invariant form (4) is naturally called the square of the
four-dimensional distance between the event point (x t ) and the origin
(0, 0, 0, 0). In view of the formal similarity to a Euclidean space, all
usual geometrical notions can now be used in the (3+1) -space. The
geometry in this space is called pseudo-Euclidean. The deviation from
Euclidean geometry is characterized by the circumstance that the
distance (4) can be zero without all (x t ) being zero. All points whose
distance from the origin is zero form a surface described by the equation
6-2 = ^ x\ =- x*+y 2 +z 2 -c 2 t 2 = 0. (6)
This surface is called the light cone, since the equation (6) describes the
propagation of a spherical light wave starting from the origin
x y = z
at the time t 0. The light cone divides the (3+l)-space into two
invariant separate domains a and b characterized by the inequalities
s 2 - x 2 +y 2 +z 2 -cW < (la)
and s 2 = x 2 +y 2 +z 2 -c 2 t 2 > 0, (76)
respectively. For events in the latter domain we can always by a Lorentz
transformation introduce a system of space-time coordinates S' in which
t' = 0, i.e. the event (a?J = (x,y,z,t) is simultaneous with the event
(0,0,0,0) in the system S'. Such a transformation is not possible for
two events in the domain (la).
An arbitrary motion of a material particle can be described by equa-
tions of the form ^ = x ^ ( t = i, 2, 3), (8)
where the X L (X^) are definite functions of the time variable x 4 . These
t H. Pomcare, Rend. Pal 21, 129 (1906).
{ H. Mmkowski, 'Raum und Zeit', Phya. ZS. 10, 104 (1909).
94 FOUR-DIMKNSIONAL FORMULATION OF IV, 33
equations represent a curve in the (3-|-l)-space which we shall call the
time track of the particle. In the case of a uniform motion the functions
x^x^) are linear and the time track is a straight line. If it goes through
the origin, it must evidently Jie entirely inside the domain (la) since
the velocity of the particle is always smaller than the velocity of light.
Then we can always introduce a system of coordinates $' such that the
#4-axis coincides with the time track of the particle. In ordinary physical
space (3-space) this means simply that we can always introduce a system
of inertia which follows the particle in its motion.
Let us now consider two arbitrary events with coordinates (xj and
(:r t ) in *S Y By transition to another system of coordinates 8' both sets of
coordinates are transformed in the same way, viz. by equations of the
form (3) with the same coefficients r\ lA . This means that the differences
(x l x t ) between these coordinates are also transformed according to
these equations, so that
I (a,--?,) 2 = I (*'-%)* (9)
I L
must be an invariant. The quantity (9) represents the square of the four-
dimensional distance between the event points (x t ) and (I*,). Since both
2 #? and 2 ** arc invariants it follows at once from (9) that
is also an invariant.
Using the expressions (3) for x\ and x\ in the right-hand side of (10)
we obtain
2 (2^ :r /)(2 a *m^m) ^ 2 (l>,/^mH^m-
t x I ' N m ' l,m x i '
According to (10) this expression must be equal to J x i^i f r a " indepen-
dent values of the variables (x t ) and (x m ). This is possible, however, only
when the coefficients satisfy the relations
2 a //^m 8 /m , (M)
/
, (0 fur I i~ m ,, rt .
uhere 8 /wl -= ] (12)
U lor I - m
is the well-known Kronecker symbol.
The relations inverse to (3) are simply
for if we use (3) in the right-hand side of (13) we get by means of (11)
2 x \<*ik =- 2 <*ti x i<*tk = 2 */(2 a 7/fA-) = 2 x i 8 M = x k>
i 1,1 i x <. } i
IV, 33
THE THEORY OF RELATIVITY
95
Using (13) in the left-hand side of (10) we then obtain by the same
argument as in the derivation of ( 1 1 )
The conditions (11) and (14) for the coefficients a ik , the so-called ortho-
gonality relations, merely express the fact that the homogeneous
Lorentz transformations represent 'rotations' of the system of co-
ordinates in (3+1) -space.
For the determinant formed by the scheme of coefficients a, A ., i.e.
(15)
we obtain, using (14) and the multiplication rule for determinants,
1
.e.
(15')
1, since a rotation may be
= 1
For proper rotations m must be equal to
performed in a continuous way, and the identical transformation x\ x %
has a scheme of coefficients whose determinant is 1. However, for
reflections where one or three ol the axes change their sign, a will have
the value 1 .
The coefficients (a iA> ) for the special Lorentz transformation (II. 24)
are given by the following scheme
y
1
\ivy/c
where y (l v 2 /c 2 )-*. It is easily verified that (16) satisfies the ortho-
gonality relations (11) and (14) and that a -- +1.
The special Lorentz transformation can also be written in the form
(16)
where we have put
COSi/r = y,
= iiy/c, tant/f ivfc.
(17)
(18)
96
FOUR-DIMENSIONAL FORMULATION OF
IV, 33
Formally the equations (17) represent a rotation in the (^ 1 ^ 4 )-plane, but
the 'angle of rotation' i/r is a purely imaginary quantity.
34. Lorentz contraction and retardation of moving clocks in
four-dimensional representation
Any event which takes place on the r-axis in the system of inertia 8
is represented by a point in the (.r^) -plane in (3+l)-space. Fig. 12,
M 1 M 2
S'\ S
Fi 12.
which is drawn as if the time variables # 4 , # 4 > an d the angle ifj were real,
gives an illustration of the Lorentz contraction and the retardation of
clocks. The lines L l and //.,, which are parallel to the x 4 -axis, represent
the time tracks of the end-points of a measuring-rod at rest on! the
ar'-axis, the rest length being / -~ x At x Al .
The length I of the measuring-rod relative to S will then be equal to
the difference in the ^-coordinates of two event points A 2 and A* which
are simultaneous (and thus have the same # 4 -value) in S, i.e.
Since all formal relations from Euclidean geometry are valid here also,
it follows from the figure that
/ --- / O sec^ = Z (l-?; 2 /c 2 ) A -, (19)
where we have used (1H). Equation (19) is the Lorentz contraction
formula. Similarly, if we consider a measuring-rod at rest on the #-axis
of the system /S, the time tracks of the end-points are given by the lines
M l and M 2 parallel to the .r 4 -axis. A consideration of the triangle B l B 2 B%
thus leads again to the equation (19).
In the same way, we may get a geometrical illustration of the clock
retardation effect. ( 'onsider a clock at rest in the system S' ; its time track
, IV, 34 THK THEORY OF RELATIVITY 97
' is given by a straight line parallel to the x^-axis, and the proper time r
is a measure of the length of this line. Now let N be this line, and let
C l C 2 be that part of the line which corresponds to the proper time r.
The corresponding time t in S is obtained as the projection of C^C^ on
the # 4 -axis. Then a consideration of the triangle C^C^C^ gives immedi-
= ry, (20)
in accordance with formula (II. 36) for the clock retardation.
On the other hand, if we have a clock which is at rest in the system $,
a consideration of the triangle D 1 D 2 D^ gives again formula (20) for the
connexion between the time /' of the system 8' and the proper time of the
clock.
In this representation the Lorentz contraction and the clock retarda-
tion effect appear as a kind of perspective shortening of measuring-rods
and time intervals. However, it must be emphasized that such a repre-
sentation is quite formal; this is also manifest from the fact that the
'angle of projection' ifi is an imaginary quantity. On the whole, the
epistemological significance of the four-dimensional representation
should not be exaggerated. In spite of the formal symmetry between the
description of space and time in the theory of relativity there is still a
fundamental physical difference between the space and time variables.
This difference is intimately connected with the difference of the
measuring instruments, clocks and measuring-rods, which are required
for the physical definition of these variables. Therefore it is not possible
by any admissible 'rotation' (3) satisfying the conditions (5) to trans-
form the time axis into a space axis Only those rotations have a physical
meaning in which the r 4 -axis remains within the domain (7 a), i.e. inside
the light cone (6).
35. Covariance of the laws of nature in four-dimensional
formulation
In spite of its purely formal character, the four-dimensional repre-
sentation has been of great significance in the development of the theory
of relativity, since it allows us to express the covariance of the laws of
nature under Lorentz transformations in a particularly simple way.
Each law of nature expresses a certain relation between physical quanti-
ties. These quantities are defined by the procedure to be applied in their
measurement. Let A, #,... be such a set of physical quantities measured
by physicists in a certain system of inertia 8. Among the quantities
A,B,... some may be so-called field quantities which are functions of
3595.60 u-
1
98 FOV H-DIMKNSIOXAL FORMULATION OF IV, 35
the space-time coordinates (x t ) in the system S. Thus a law of nature
can be expressed by one or several equations of the form
,...,** **} = <>,
<>*, 0** I
(21)
where F is a function of the quantities A, Z?,... and possibly of their
derivatives of arbitrarily high order with respect to the space-time
coordinates.
Physicists in another system of inertia 8' will now by means of the
same measuring methods generally find other values A' ', B' ,... for the
physical quantities mentioned, and if, for example, A is a field variable,
A' will generally be a different function of the coordinates (jr' t ) than A is
of (x t ). Thus the field variables as functions of the space-time coordinates
are generally not form -invariant.
However, the physical law expressed by (21) in S can in S' be expressed
by equations of the form
',H',...^ A ' t , BB ;,. .)=,0, (21')
, ^'L I
where the function F in (21') on account of the special principle of rela-
tivity must be the same function of the arguments (A', #',...) as is the
function F of (A, B,...) in (21), i.e. any relation between physical quan-
tities must be expressed by means of form -invariant or covariant
equations.
When the problem arose of expressing the fundamental equations in
the theories of electrodynamics and elasticity in a form independent of
the Cartesian system of coordinates applied in the description, the
three-dimensional vector and tensor calculus was invented. Since the
Lorentz transformations represent rotations in (3 +- l)-space, it is there-
fore natural to attempt to meet the requirement of covariance of the
laws of nature under these transformations by a generalization of the
three-dimensional vectors and tensors to four dimensions and to write
the fundamental equations in the form of four-dimensional tensor
equations.
As we shall see in the following chapters, this was possible for all
fundamental equations in classical macroscopic physics, and for some
time it was even believed that all laws of nature could be written in
tensor form. With Dirac's quantum mechanical theory of the electron*)*
it became clear, however, that it is necessary for certain physical systems
to deal with other quantities besides tensors, the so-called spinors j which
t P. A. M. Dime, The Principles of Quantum Mechanic* (3rd ol., Oxford 19-47),
chap. xi.
IV, 35 THE THEORY OK KKLATIVITY 90
have quite different transformation properties, but nevertheless satisfy
co variant differential equations of the type of (21), (21').
36. The four-dimensional line element or interval. Four-
vectors
Consider two neighbouring points P and P' in (3+l)-space with the
coordinates (xj and (x^dx^ in an arbitrary system of coordinates S.
On account of (9) the square of the four-dimensional distance between
these points is given by 2
This expression for the line element or the interval defines the geometry
in (S-fl)-space. The infinitesimal line connecting P and P' is now the
prototype of a vector, just as in a three-dimensional space. This vector is
defined by its four 'components' (dx { ) relative to an arbitrary system of
coordinates S. By a rotation of the system of coordinates leading to the
system S' these components transform like the coordinates, i.e. we have
dx 'i = 2 ik <fak- ( 23 )
Now, a four-vector is quite generally defined as a quantity which relative
to every system of coordinates has four components (a t -) which are trans-
formed in the same way as coordinates (x^. Thus, to the rotation (3), (13)
of the system of coordinates correspond the transformation equations
***> a k = 2 X-* ( 24 )
for the components of a four- vector.
In analogy with (4) it then follows from (11) and (14) that the 'square
of the magnitude of the four-vector', or the norm of the vector
? = !? (25)
is an invariant.
According as the invariant (25) is less than zero, equal to zero, or larger
than zero, we speak of a time-like vector, a zero vector, and a space-like,
vector, respectively.
For the infinitesimal vector (dxj connecting the two neighbouring
events P and P', (25) becomes identical with (22), viz.
cfo* = dxt = d(7 2 -c 2 eft 2 , (26)
i
where do is the spatial distance between the two space points in physical
space in which the two events occur, while dt is the difference in the time
of occurrence of the events. The light cone with P as centre is defined by
the equation ds 2 = 0. If P' lies on the light cone, the two events can be
100 FOUR-DIMENSIONAL FORMULATION OF IV, 36
connected by means of a light signal. If the vector (dx^ is time-like,
P' lies inside the light cone, while it lies outside the cone when (dxj is
space-like.
From two four- vectors with the components (a t ) and (6 t ) can be formed
a new vector with the components (a^-f-ftj.
Since, furthermore, the invariant square of the magnitude of this
vector can be written
(27)
I ( t +M 2 - 2 ?
i ii
it follows immediately that the quantity
is also an invariant in analogy with (10). The quantity (27) is called the
scalar product of the two vectors. Two vectors are said to be orthogonal
to each other when their scalar product is zero.
The first three components of a four- vector behave under spatial rota-
tions like the components of an ordinary spatial vector a. Therefore, in
any system of inertia we can split a four- vector into a spatial part and a
temporal part (,) - (a,a 4 ), (28)
but this splitting is of course not invariant under Lorentz transfor-
mations. As the quantities (a, 4 ) transform in the same way as
we have obviously for an arbitrary Lorentz transformation without
rotation, according to (II. 25'),
v
a' + - 2
V ^ "'" ' ' (29)
When the vector is time-like, i.e.
2>?= |a|2-|a 4 |2<0, (30o)
<
we can always introduce a system of coordinates S' such that the spatial
vector a! in this system. We haye simply to choose the time axis
in 8' in the direction of the four- vector (a t ). If, however, the vector is
space-like, ^a* = |a *-\atf > 0, (306)
I
we can find a system S' in which a 4 = 0. This obviously means that the
IV, 36 THE THEORY OF RELATIVITY 101
new time axis is perpendicular to the vector (a^), for, if (b T ) denotes a
vector in the direction of the new time axis, fc[ = for i = 1, 2, 3, i.e.
a t 6 t = a;6J = 0.
37. Four -velocity and acceleration. Wave -number vector. Four-
ray velocity
Let us now consider the motion of a material particle and the corre-
sponding time track (8) in the (3+ 1 )-space. We can also use a parameter
representation for the time track
*t = ^() ('= 1,2,3,4) (31)
with the length s of the curve as parameter. Two neighbouring event
points on this curve are connected by an infinitesimal four- vector (dxj
with the length ds given by (22) or (26). Since the velocity of a particle
is always less than c, it follows from (26) that ds 2 is negative everywhere
on the curve. It is therefore convenient to introduce instead of s a new
real parameter r defined by
_
o - 1CT .
Using (32) in (26) we obtain
c 2 dr 2 = da 2 -c 2 dt 2 , (33)
and, since the velocity of the particle u is equal to dv/dt, we get
dr = (Iu 2 /c*)*dt. (34)
Thus r is identical with the proper time of the particle, i.e. with the time
measured by a standard clock which follows the motion of the particle.
The proper time of a particle is a measure of the length of the time track.
The time track corresponding to a uniform straight motion is repre-
sented by a straight line as, for example, the line N t in Fig. 12, which
connects the two events A and B. The length of this line is given by
= T > = <'--'VHJ)' (35)
where u^ is the velocity in the uniform motion and t B t A is the difference
in time between the two events A and B. If we consider another arbitrary
time track N 29 connecting the same events A and B, it obviously repre-
sents a non-uniform motion. The length of this curve is then, according
to (34), given by
(36)
Since the expressions (35) and (36) hold in any system of coordinates
we can, for example, perform the calculation in a system of coordinates
102 FOUR-DIMENSIONAL FORMULATION OF IV, 37
whose time axis is parallel to the straight time track N v This means only
that we introduce a system of inertia following the uniformly moving
particle. In this system of coordinates u l -= 0, while the velocity u in
the non-uniform motion corresponding to K 2 cannot possibly be zero along
the whole track. Therefore we must have
*:-< s . 1 . (37)
1C 1C
The straight motion is thus characterized by the fact that the length of
the time track in this case has a stationary value, viz. a maximum, com-
pared with all other possible motions connecting the same two events.
Since the coordinate increments (r/.rj on the time track of a particle
are the components of a four- vector and dr is an invariant,
will also be components of a four- vector which is called the four -velocity.
From (38) and (34) we see that the components of U L are
c u
where u ~ (40)
is the usual three-dimensional velocity vector. The four-velocity is a
four-vector which lies in the direction of the tangent of the time track
and the norm of this vector has the constant value c 2 , for from (39)
one obtains .,
Thus the four- velocity is a time-like vector of constant magnitude and
by differentiation of (4 1) we get
If we apply the equation (29) to the four-vector U t given by (39) we
come back to the transformation equation (II. 55') for velocities and to
the equation inverse to (II. 56).
The four-vector
IV, 37 THE THEORY OF RELATIVITY 103
is called the four-acceleration and from (39) we get for its components
2 3
(42)
du r/ 2 x
where a = - rf/ - = -
is the usual three-dimensional acceleration of the particle. According
to (41') the vectors U l and A l are orthogonal to each other. In the rest
system 8' of the particle the components of A t are
^'.-(a',0). (42')
Another example of a four-vector is given by the wave number vector
(cr,) of a plane monochromatic wave The invariant phase F of the wave
in an arbitrary system of coordinates 8 can, on account of (II. 68), be
written
Here, n denotes a unit three- vector in the direction of the wave normal,
v is the frequency ?r is the phase velocity, and A is the wave-length of
the wave. As the phase is an invariant we have, according to (3),
, ,A
and, since this equation must hold for arbitrary x k , we get
which shows that (a,) is a four- vector. Equations (44), or their equivalent
(29), yield directly formulae (70), (71), arid (72) deduced in 23 for the
transformation of the characteristics of a plane wave.
In 24, Chapter II, we saw that the ray velocity in a medium with
refractive index n > 1 transforms like the velocity of a particle with the
velocity u' = w' - cjn in the rest system of the medium. In analogy
with (39) we can therefore define a four-ray velocity vector (f/J with the
components , e
where u is the magnitude of the ray velocity and e is a unit three-vector
in the direction of the ray.
In the rest system $' of the refractive medium where e' n' and
w' u' = c/n, we have
cri icn
(46)
104 FOUR-DIMENSIONAL FORMULATION OF IV, 37
In another arbitrary system of inertia S we then have U k -- ^ U\oL tk .
From (46) it follows that
SX ", = 5X "', ^ <>. (47)
> t
i.e. the two four-vectors o- t and f ^ are orthogonal to each other. Inserting
(43) and (45) we obtain
or w; (u.n). (47')
(47) thus expresses simply that the phase velocity is equal to the pro-
jection of the ray velocity in the direction of the wave normal in any
system of inertia.
While the square of the magnitude of (U t ) is constant, equal to c 2 ,
we have
Thus, for a light wave in empty space the phase wave vector is a zero
vector.
38. Four -momentum. Four-force. Fundamental equations of
point mechanics in four -dimensional vector form
In 28, formulae (32) and (37), we have seen that momentum and
energy (p, K[c) of a material particle transform in the same way as the
space-time coordinates (x,r). Therefore the four quantities
Pi-- (p,iM/c) (49)
are the components of a four- vector, the four-momentum vector. When
we use (III. 23) and (III. 31) in (49) we see that the four-momentum
p l is proportional to the four-velocity (39) and the factor of proportion-
ality is equal to the rest mass m of the particle, i.e.
p, -m^. (50)
On account of (50), (49), and (41) the norm of this vector is
IP? = P*-- -* - 2 2 f ; f -- -8<- a (< r >i)
i < /
in accordance with (III. 35) The four-momentum vector is thus a time-
like vector.
Since p t in (50) and a, in (43) are transformed in the same way, it is
possible in ari invariant way to adjoin a plane wave with the wave
number vector cr l to a free particle with four-momentum p^ such that
/, =- K, (52)
IV, 38 THE THEORY OF RELATIVITY 105
where h is an invariant constant. If h is chosen equal to Planck's con-
stant, the adjoined wave is the de Broglie wave| of the particle. The
phase velocity w of the de Broglie wave is, according to (43), (52), and
(III. 36), connected with the velocity u of the particle by the equation
- l -_- *l a J. = * |P| =.- p - ^ U . (52')
w cor 4 cp 4 E r 2 '
The theorems of conservation of momentum and energy in a collision
between particles can now be comprised in the equation
where p^ and p t are the sum of the four-momentum vectors of the par-
ticles before and after the collision, respectively. The first three equations
(i --- 1,2,3) represent the theorem of conservation of momentum while the
fourth equation, corresponding to i -- 4, expresses the conservation of
energy.
In (53) these theorems are expressed in a form in which the co variance
under Lorentz transformations is obvious, for two four-vectors whose
components are equal in one system of coordinates will obviously have
equal components in any system of coordinates.
As mentioned in the conclusion of 28, the quantities (F^ 7 , (Fy.u))
transform in the same way as (p, E), i.e. the four quantities
are the components of a four- vector, Mirikowski's four-force. \ The
fundamental equations of mechanics (III. 42) can thus be written in the
following co variant four- vector form
or, since m is constant in time,
ro ^ - F t . (56)
dr
The first three equations are the equations of motion, and the fourth
equation expresses the theorem of conservation of energy. Conversely,
from the validity of (55) in every system of inertia it follows at once that
f\ is a four-vector, for F t must then transform in the same way as the
left-hand side of (55) which is obviously a four- vector, since r is invariant.
t Op cit , chap, u, p. 58
t H. Mmkowski, k Das Relativitatsprmzip', Ann. d. Phys. 47, 927 (1915).
106 FOUR-DIMENSIONAL FORMULATION OF IV, 38
From (41') and (56) it is seen that the vectors L\ and F l are orthogonal
to each other, viz. y p , r .._ n ir^\
* i i * \ '
This property of F l is closely connected with the constancy in time of
the rest mass. From (55), (41), and (41') we obtain
y p jj __ X^ [ j fl( m o *-i) __ ^ jjz ^ m o j m ^C jj d^>i __ ^2 _Q
i l l J-* l dr <^-4 l dr ' ^-4 l dr dr
i i <
(58)
Equation (57) thus expresses the fact that the proper mass m Q is con-
served.
Sometimes we are concerned with systems in which the external
forces produce a change in the proper mass of the particle. This is the
case, for instance, in an electrically conducting substance under the
influence of electromagnetic forces, since the Joule heat energy produced
in the body will contribute to the proper mass. If in such a case we wish
to maintain the equations (55), we must in the expression (54) for F x
replace F . u = A by the total effect. While F v F 2 , F 3 remain unchanged,
F* will be defined by
*\ - ^
where A ^ (F.u) '(60)
is the mechanical work performed per unit of time, while Q is the amount
of heat or non-mechanical energy developed per unit of time in the body.
The fourth equation (55) which, by means of (49), (59), and (34) can be
written ,
=^+. (61)
thus again expresses the conservation of energy. As the equations (55)
must hold in every system of inertia, and since the left-hand side is a
four-vector, the generalized four-force must also be a four-vector. But
now (F L ) is no longer orthogonal to (f/,). Instead, we have
= _ _ 9 _ (62)
' ^ >
Since the left-hand side of this equation is an invariant,
must thus be an invariant. It is equal to the amount of heat developed
in the rest system per unit time.
IV, 38 THE THEORY OF RELATIVITY 107
In a physical system of this kind the rest mass of the particle is no
longer conserved. From (58), (62), and (63) we get
(tWln Vj/ /f*A\
-7- 1 = - 2 ( 64
dr c*
Since r is identical with the time in the rest system, (64) simply means
that the amount of heat developed in the rest system has an inertial mass
corresponding to Einstein's equation (III. 74).
While the equation (55) thus holds quite generally, (56) is valid only
when the. rest mass is conserved, i.e. when ^ t F l U l == 0.
The equation (55) may also be written
m dU ,,dm 0u = f
dr dr
or, on account of (58),
7 1 1 O 1' \ f
dr c*
Thus, when ^ F l f/, ^ 0, i.e. when the proper mass is not conserved, a force
will in general he necessary in order to maintain a uniform motion of the
particle; for dUJdr only if
I tf IT \ (JO
-Jd u i = v Uif (65)
In the rest system this means F = 0, but in every other system of inertia
we have
From (63) we get at the same time the transformation properties of the
amount of heat Q conveyed to a system in a process of that kind. Multi-
plying (63) by Ar^l u 2 /c*) we obtain, using (34),
= Q A is the total amount of heat conveyed to the system during
an interval A, while A$ = Q Q AT is the corresponding quantity
measured in the rest system. Thus we have
In Chapter VII we shall see that the equation (66) also holds for any
amount of heat conveyed to a system in a thermodynamical process.
108 FOUR- DIMENSIONAL FORMULATION OF IV, 39
39. Tensors of rank 2
In the preceding chapters we have seen that the covariance of the
fundamental equations of mechanics under Lorentz transformations can
be expressed in an especially elegant way by writing them in four-
dimensional vector form. To obtain a similar geometrical representation
of electrodynamics, for instance, it is necessary to introduce four-
dimensional tensors also.
By a tensor of second rank in (3+ l)-space we mean a quantity which
has 4 2 components (t lk ) relative to an arbitrary system of coordinates S
and whose components (t\ k ) in another arbitrary system 8 f are con-
nected with the components (t lk ) by means of the equations
( 67 )
where the a lk are the same coefficients as in equation (3) which defines
the transition from S to S'.
For the sake of simplicity we have here omitted the sign of summation,
substituting it in the following by the convention that an expression in
which a Latin index, like / or m in (67), appears twice shall be summed
over this index from 1 to 4. Free indices like i and k in (67) can assume
independently the values 1, 2, 3, 4. Equation (25), for example, will thus
in the future be written a l a l = a\a\. If an index can assume only the
values 1, 2, 3 it will be denoted by a Greek letter, and if it appears twice
in an expression it is implied that we shall sum over this index from 1 to 3.
The square of the magnitude of a spatial vector will thus be written
a| 2 - a t a t .
The definition (67) of a four-tensor is a direct generalization of an
ordinary spatial tensor whose components by rotations in physical space
transform according to the equations
where OC LK are the coefficients in the orthogonal transformation repre-
senting the rotation.
The sum of the diagonal elements of a tensor of second rank is an
invariant, for from (67) and (11) we obtain
In the same way it is seen by means of the orthogonality relations (11),
(14) that the quantity . . __ ., .,
l ik l ik hmhm
is an invariant.
IV, 39 THE THEORY OF RELATIVITY . 109
If a Lk represents ,a simple rotation in the three-dimensional physical
space, we have A T
r = = * = 1
and (67) is reduced to
C = aiA<V*A/i *'i4 = a <A^A4 ^ (72)
^4* == a *^4/i ^44 ~ ~ ^44 '
(72) shows that the spatial part of a four-tensor by spatial rotations
behaves like an ordinary three-dimensional tensor. Furthermore, the
numbers t4 and t K separately form the components of a spatial vector,
while / 44 is an invariant for purely spatial rotations.
From a vector a t and a tensor t lk a new vector can be formed whose
components in every system of coordinates are given by the equation
*, - t ik a k9 (73)
for from (24), (67), and (II) we obtain
If a lk and b lk are the components of two tensors, fl^+^
viously also be the components of a tensor which is called the sum of the two
tensors. Further, if t lk is a tensor, t tk = t kl is also a tensor, viz. the trans-
posed tensor. When a t and b t denote the components of two vectors
which thus are transformed according to (24), the quantities
',*--- A ( 75 )
will obviously be transformed according to (67). The tensor of second
rank, defined by (75), is called the direct product of the vectors a l and b k .
Also the quantities , , , . ,-.
M *ik = a i b k~ a k b i = *ki> ( 76 )
which denote the difference between the tensor (75) and its transposed,
thus represent a tensor. A tensor like (76), satisfying the equation
for all values of the indices i and /*, is called antisymmetiical. Analo-
gously, a tensor satisfying the equations
',* = '*i-i,A ( 78 )
is called symmetrical.
Since both sides of the equations (77) and (78) transform like tensors,
these equations must hold in every system of coordinates if they hold in
one system. Symmetry or antisymmetry of a tensor is thus an invariant
110 FOUR. DIMENSIONAL FORMULATION OF IV, 39
property. In an antisymmetrical tensor all diagonal elements are zero,
for if we put i = k in (77) (without summing), we obtain
^ - -*,* = (79)
for any i.
An antisymmetrical four-tensor of second order F ik = F kl has only
six independent components. Putting
H tK - F iK9 E L = iF L , = -tT 4l , (80)
H IK and E L will, according to (72), in spatial rotations (71) behave like
components of an antisymmetrical spatial tensor and an ordinary space
vector E, respectively.
Moreover, putting
//! = #*, #,= //, #3 =#12, (80')
we obtain the following transformation equation for H t and E t in the case
of a special Lorentz transformation (16),
\ = E 19 E' 2
\
'
where y = (1 v 2 /c 2 )-*-. The corresponding reciprocal equations are
obtained by interchanging the primed and unprimed quantities and
replacing v by v. With v = (v, 0, 0) the latter equations may be
comprised in the vector formulae
E = yE'-f-
: (81')
H = yH' + -i
v 2
and in this form they are valid for any Lorentz transformation without
rotation of the spatial axes.
40. Angular momentum and moment of force in four -dimen-
sional representation
Let (#J = (x,ict) be the space-time coordinates of an event point on
the time track of a material particle, and let (p t ) {p, i(E/c)} be the four-
momentum of the particle. According to (76), from these two four- vectors
we can form an antisymmetrical tensor
J^f __ ^ p y. rp (82)
The spatial part of this tensor is an antisymmetrical space tensor, the
IV, 40 THK THKORY OF RELATIVITY 111
angular momentum tensor M LK , whose components are connected with
the angular momentum vector
M xx p (83)
bv the equations
M = (M f , M,,, M s ) = (M n , M 3l , M n ). (84)
In the same way we can from (x,) and from the Minkowski four-force
(FJ form the tensor ^ ^ ^ ^_^ ^ (85)
The spatial part of this tensor gives, analogously to (83), (84), the
moment of the Minkowski force (III. 41) relative to the origin.
By means of the fundamental equations of mechanics (55) we obtain,
using (38),
(86)
or, by means of (50) and (85),
dM lk
The spatial part of this equation contains the angular momentum
theorem , M
^ = (xxF). (86')
The Kronecker symbol defined by (12) represents a tensor of second
rank of especially simple character. Consider a tensor whose components
in 8 are equal to S, A ; on account of (67) and (14) its components in $' are
'
This tensor has thus the same constant components in every system of
coordinates and it is the only tensor whose components remain un-
changed by a transition to another system of coordinates.
^~~ ^
41.) Tensors of arbitrary rank
^ /
In analogy with (67) a tensor of third rank in (3-fl)-space is now
defined as a quantity with 4 3 components t lkl which transform according
to the equations
C/ = a 7m (X-kn <*/;> 4>/^>' ^kl = *row;> a mt <*nk a j)l' ( 88 )
Thus every index transforms separately according to the same law as
for a four-vector. A $owr-vw^
rank. In the same sense an aiajUA^aJe.nsor^) zero rank.
A tensor of rank n is then a quantity t ikl ...
each of whicTT transforms separately according to the law (24),
112 FOUR-DIMENSIONAL FORMULATION OF IV, 41
characteristic of a vector. If in a tensor of rank n two indices, for example
k and i, are put equal to each other, we obtain after summing over this
index a tensor t lllm of rank (n 2). This is a direct consequence of the
transformation equations for tensors in connexion with the orthogon-
ality relations (11), (14). Such a process by which from a tensor of rank
n a tensor of rank (n 2) is formed is called contraction. Equation (69),
in which a tensor of rank zero has been formed from a tensor of rank
2, represents a special case of such a contraction.
By addition (or subtraction) of corresponding components of two
tensors of rank n we naturally obtain a new tensor of rank n. On the
other hand, it has no covariant meaning to add two tensors of different
rank. However, we can always form the direct product of two tensors of
ranks n and m, respectively, by forming all possible products of the
components of these tensors. Hereby we obtain a new tensor of rank
(n-\-m). The equation (75) obviously represents a special case of this
general theorem, since the tensor t lk = a l b k of rank 2 is the direct
product of the two tensors of first rank a l and b k . By subsequent con-
traction of the tensor d l b k we obtain a tensor of rank zero, viz. the
invariant (27).
Equation (73) also represents a special case of a combination of both
operations: direct multiplication and contraction. Primarily, a tensor
(t ik .a t ) of rank 3 is formed by direct multiplication of the two tensors
t lk and a t . By contraction we then obtain the tensor b l t lk a k of first
rank. In the same way, equation (70) can be regarded as a result of a
direct multiplication of t lk by itself succeeded by two contractions.
42. Pseudo- tensors
In three-dimensional vector calculus one introduces besides the
ordinary (polar) vectors with the transformation law
a 't = <*** ( 89 )
so-called axial vectors which transform according to the equations
a\ - ococ lK a^ (90)
where a = \oc iK \ is the transformation determinant. For proper rotations
we have a = 1, and an axial vector transforms like a polar vector.
However, by reflections in which one or three axes change their signs we
have a 1 : thus, for instance, by a reflection at the origin in which
x[ - -x t , (91)
the components of an axial vector are unchanged, while the components
IV, 42 THK THEORY OF RELATIVITY 113
of a polar vector change their signs. An example of an axial vector is the
vector product c of two polar vectors a and b with the components
c --- (c^Cg.rg) = ((I 2 b^-a 3 b 2 ,a 3 b l a 1 b^a 1 b 2 ~a 2 b l ). (92)
By a reflection a[ = t , b[ = fo t , we have obviously c[ C L . Another
well-known example of an axial vector is the magnetic field vector H.
The generalization of the notion of an axial vector to tensors of higher
rank and to four dimensions is obvious. These quantities are called
pseudo-tensors. They transform like tensors, except that they are also
multiplied by the transformation determinant at \oc lk \ defined by (15).
Thus, a pseudo-tensor of rank 2 is a quantity with 4 2 components in every
system of coordinates with the transformation law
t\ k ---- oux lt ot km ti m . (93)
From this definition we get at once the following rules. The sum of two
pseudo-tensors of the same rank is again a pseudo-tensor of equal rank.
The direct product of a pseudo-tensor and a tensor is a pseudo-tensor
with a rank equal to the sum of the ranks of the two factors in the product.
The direct product of two pseudo-tensors is a tensor. The operation
of contraction can be performed with pseudo-tensors in the same way
as with tensors, thus leading to a pseudo-tensor whose rank is diminished
by 2.
43. The Levi-Civita symbol
Like the Kroiiecker symbol, which was shown in 40 to be a tensor
with constant components in every coordinate system, the Levi-
Civita symbol is a pseudo-tensor with the same property. In four-
dimensional space this pseudo-tensor is of rank 4. The Levi-Civita
symbol is defined as a quantity S, A/m which is antisymmetric in all four
indices. Thus, the only won -vanishing components of 8 lkhn are those for
which all four indices are, different and they are equal to + 1 or 1 according
as (i, A*, /, m) is an eren or an odd permutation of (1,2, 3, 4). Now consider
a pseudo-tensor which in 8 has the components 8 <A/m . In another system
S' its components are then
mr 8 w/f , r . (94)
Since all symmetry properties are conserved by the transformation,
8' tUm is also antisymmetric in all indices and we need only calculate the
component with (i, A% /, m) = (1,2,3, 4) for which -we get
&'m4 = *i,,2,,3v*4A,,,,. (5)
3595.60 i
114 FOUR-DIMENSIONAL FORMULATION OF IV, 43
From the definition of 8 lklm it follows that
thus, by means of (95) and (15'),
8 1234 = <* 2 == 1 = S 1234-
From the symmetry properties of 8 f lklm and 8 lWm it then follows that
8 !*/m = 8 tA/m ( 96 )
for all values of the indices (i, i% /, m), which shows that the Levi-Civita
symbol is a pseudo-tensor with the same constant components in every
coordinate system.
In three-dimensional space, the Levi-Civita symbol is a quantity
8 lK x antisymmetric in all three indices. S 123 is equal to I, the other non-
vanishing components following from this by the symmetry rules. It is
shown in the same way as before that 8 llf ^ is a three-dimensional pseudo-
tensor.
44. Dual tensors
By means of the Levi-Civita symbol we can associate an antisym-
metric three-tensor H LK with a pseudo-vector (axial vector) H by
#i=4SiKAtf*A, (97)
i.e. H - (H 19 # 2 , ff 8 ) - (# 23 , #31, # 12 ). (98)
The quantities H l defined by (80') are thus the components of an axial
vector dual to the tensor H IK .
If H IK is of the form
", = A,-aA= I 1 I' , (99)
where a t and b K are two vectors, the corresponding axial vector
^ = S^a^A (100)
is the vector product o = a x b. This tensor or its dual axial vector
represents the parallelogram formed by the vectors a and b, their com-
ponents being equal to the projections of the parallelogram on the three
coordinate planes. The area a of the parallelogram is given by the
equation 2 i /im\
M <r 2 = ^1= 4< T K< r c- (101)
The pseudo-vector <r t is perpendicular to the parallelogram, for from
(100) we get
*ti = t SHCA^A - and a i b l = (102)
IV, 44
THE THEORY OF RELATIVITY
115
on account of the antisymmetry of the Levi-Civita symbol. The vector
dual to the tensor ~ ~
= fa K __ fa;
IK dx t 'dx K
is similarly the axial vector curl a.
Three vectors a, b, and c define a parallelepiped which by analogy
with (99) is represented by an antisymmetrical tensor
(103)
By means of the Levi-Civita symbol we can associate this tensor with a
pseudo-tensor of rank zero, i.e. with a pseudo-invariant
v = ~
ftl
a 2 ^2
(104)
F represents the volume of the parallelepiped. It is invariant for
proper rotations, but changes sign by reflections. We obviously have
F 2 = V
' '
(105)
If a, b, c are infinitesimal vectors lying in the directions of the x^,
x 2 - y and x 3 -axes, respectively, for instance
a = (efcq, 0, 0), b = (0,rf# 2 , 0), c = (0, 0,d^ 3 ), (106)
the corresponding volume element is, by (104) and (106),
dV = dx^x^dx^. (107)
It is a pseudo -in variant.
In (3+l)-space we can associate an antisymmetrical tensor of rank
n < 4 with a pseudo-tensor of rank (4 r?) by means of the Levi-Civita
symbol 8 iklm . Thus the dual pseudo-tensor FJ k to an antisymmetrical
tensor F lk is defined by the equations
F* k = ^u m f lm , (108)
i.e.
1
= ^
23'
= ^.
(109)
Introducing the quantities E and H denned by (80) and (80') we see that
the dual pseudo-tensor Ff k is obtained from F lk by the substitution
116 FOUR-DIMENSIONAL FORMULATION OF IV, 44
E -> H, H -> E. The equations (81) and (81') are unchanged by this
substitution, in accordance with the fact that the Lorentz transformation
without rotation is a transformation with a = 1, so that F* k in this
case transforms in the same way as F lk .
A tensor a^ k of the special form
where a l and b k are two vectors, represents the two-dimensional 'parallelo-
gram' defined by the vectors a t and b l by analogy with (99). The dual
tensor
,
*. = -
is orthogonal to the vectors a l and b k and to the tensor a lk -
^-^A-<4<^-<>. (HI)
The area a of the parallelogram is defined by
by analogy with (101).
Likewise, to an antisymmetrical tensor of rank 3 corresponds a dual
pseudo-tensor of rank 1, i e. a pseudo-vector. If the tensor is of the form
(113)
*/ '' "
where b^ c, are three independent vectors, the dual pseudo-vector is
F - l 3 V - -
where (iklm) is an even permutation of (1234).
V lkl and V l represent the three-dimensional parallelepiped defined by
the vectors a p ft t , c t -. ^ is orthogonal to this space, for we get from (114)
JX = FA = F t c t = 0. (116)
The volume V of the parallelepiped is given by the length of the pseudo-
vector V l ,
F*=-F t F t = lF Wm F fc/m . (117)
IV, 44
THE THEORY OF RELATIVITY
117
Finally, the dual pseudo-tensor of an antisymmetrical tensor of rank 4
is a pseudo -invariant. If the tensor is of the form
a m b m C m d m
(118)
the dual pseudo-invariant is
v - l 8
- - S, ^
__ I g a b c d __ I
- iklm i 'k l f m ~ ^
(119)
2 represents the volume of the parallelepiped defined by the vectors
abcd We have ,
(120)
If a p b l , c e , d t are infinitesimal vectors lying in the directions of the co-
ordinate axes and of lengths dx } , dx 2 , r/a* 3 , ^ 4 , respectively, the corre-
sponding four-dimensional volume element is
(121)
which is thus a pseudo-invariant.
45. Infinitesimal Lorentz transformations. Lorentz transforma-
tions without rotation
An infinitesimal homogeneous linear transformation (x z ) -> (x\) has
the form , /s i \ /ioo\
x ^ - x l + lk x k = (B lk + lle )x k9 (122)
where the * lk are infinitesimal quantities. For a Lorentz transformation
we get, using (122) in (10) and neglecting terms of second order
in *>
Since this equation must hold for all values of x l and x i we must have
t*=-^ t . ' ( 123 )
This condition is equivalent to the orthogonality relations (11), (14) in
the case of an infinitesimal transformation.
Now consider an arbitrary Lorentz transformation (3) connecting the
space-time coordinates of two systems S and S'. Let v (v x , v u , v z ) be
118 FOUR-DIMENSIONAL FORMULATION OF IV, 45
the velocity of 8' relative to S. The corresponding four-velocity V i is then
by (39) F ( =(yv,yic) (y = (l-/c)-').
The components of this four- vector relative to S' are
F;= (0,0,0, ic),
since J^ is the four- velocity of a point at rest in S', which means that the
vector V i lies in the direction of the a^-axis. From the transformation
equations of a four- vector we then get
V k = V\oi lk = ICOL^. (124)
Similarly, if e( l \ e( z \ ^ 3) are unit vectors in the directions of the x\-, #' 2 ~>
#3-axes, respectively, we have
#' = 8*. (t=l,2,3)
and (.) __ d)> ___ * __ v
6 <o ~~(o ^ ~~ U ^ ~ ]
In the case of a Lorentz transformation without rotation the co-
efficients oL lk follow from (II. 27):
V y , -.v
--i(y-i)
^ (y-1)
" i
i(y-i)
(125)
C C C C
In (3+l)-space it represents a rotation in the two-dimensional plane
defined by the time axes of S' and S.
If v x , v u , v z are infinitesimal quantities, we have, neglecting second-
order terms in these quantities,
and (125) reduces to
a = & lk + lk with IK - 0, l4 = 4l = ^>, 44 - 0. (126)
I/, t& I tfc lie > (4 41 c 44 V ;
46. Successive Lorentz transformations
Let *; = * |A * A , 8-+S', xl^oc^xl S'^S" (127)
be two successive Lorentz transformations. The resultant trans-
formation r"~-U',v W H9^
X t (a^oiiLJXL V 1 - 8 ;
IV, 46 THE THEORY OF RELATIVITY 119
is, of course, again a Lorentz transformation, i.e. the coefficients
satisfy the same orthogonality relations (11) and (14) as oc lk and oL lk .
However, (128) will not in general represent a Lorentz transformation
without rotation even if this is the case for the two transformations ( 1 27),
i.e. Qt!' lk will not take the form (125) even if ot ik and oL ik are of this form.
This will be the case only if the three time axes in S, $', and S" are lying
in the same plane.
In the special case where the transformation from S r to S" is an in-
finitesimal Lorentz transformation without rotation, we have, according
to (126),
*ik = 8 t* + 'ifc *IK = 't4 = ~ 4t = - 44 = > ( 129 )
c
V( being the four- velocity of S" relative to S'. Hence,
<*lk = ( 8 t/+ </)/* = *ik+*il<xik' (1.30)
Let x i = f t (r)
represent the time track of a particle in arbitrary motion in S, r being
the proper time of the particle. We shall now try to determine the
successive rest systems of the particle such that two consecutive rest
systems at any time have the same orientation of the spatial axes. Let
S' and S" in (127) be momentary rest systems of the particle at the times
r and T-fdr, respectively. The four- velocity of S' relative to S is then
^(T) = ^=/,(T). (131)
Similarly, the four-velocity of S" relative to S is
V { = tf t (T)+dtf t (T) = U t (T)+tTt dr = /<(T)+/(T) dr. (132)
The components of these two four-velocities in S' are
7;(T) = tt t^ = (0,0,0,tc), (133)
V't = U:+dU' t = a ik (U k +dU k ] = U'i+a^ dU k . (134)
Since the transformation from S f to S" was supposed to be an infinitesi-
mal Lorentz transformation without rotation, the coefficients oL n ik in the
transformation from S to S" are obtained from (130) and (129), V\ in
(129) being given by (134). Now, obviously, we have
C
for, on account of (133), this is seen to be identical with the t' ik defined in
(129).
120 KOUR-DIMENSIONAL FORMULATION OF IV, 46
Thus, from (130), (135), and (134)
ij\du'-ir l dv\ i IT . ... .. JTr .
,*-* = - -ir~~- - <% = -,a( r7 k dUt-Ui dU k ).
The coefficients oc lk can now be regarded as functions <y. lk (r) of r, a" lk being
then equal to oi lk (r~\-dr). Hence we get the following differential equa-
tions for the functions a lk (r):
(136)
.., U^lJi. C/T. C/y /, v
with ri %k = -J-A_JL_J . (137)
For later use we note that the coefficients ot lk and rj lk satisfy the relations
.=
Kl ^l
( J 33 X
on account of (14), (41), (41') and (133).
The equations (136) determine the transformation from the fixed
system S to the momentary rest system #' S'(T) with coordinates
x' thus we have ^ ' / \ /ion\
1 ^ = x k oi kl (r), (139)
or, if we make a continuous displacement of the origin in S' such that the
particle is always lying at the origin of the rest systems $'( T )>
). (140)
, Let us now attach a space vector e'(r) of unit length to the particle
considered in such a way that the components e' with respect to the
spatial axes of S'(T) have the same values at all times. e'(r) may, for
instance, have the direction of one of the space axes in A$> T/ . We thus have
at any time a displacement of e' without change of orientation. In
(3+l)-space this vector is represented by a space-like four- vector with
components given by e ; -= (e', 0) (141)
in 8'. Its components in S are given by
^(r) = fa^r) = e'^T). (142)
e l is orthogonal to U 19 since
e t t7 <= =e;C/; = (143)
IV, 46 THE THEORY OF RELATIVITY 121
on account of (133) and (141). From (142), (136), (137), and (143) we
now have , , ,
_
~ k
dr ~ dr
i.e. ^'. (144)
dr c 2
If S'(r) coincides with S for r = we have
Cf = < for T =
and the velocity of the particle is zero at that moment. At a later time
we have in general e r ^ e\, and even if the velocity of the particle becomes
zero again, we shall in general have
e r = (e,0) =<== (e',0)
at that time. This means that the components of the unit vector con-
sidered are different in 8 and AS", in accordance with the fact that the
vector has performed a Thomas precession relative to S (see 22).
47. Successive rest systems of a particle in arbitrary rectilinear
motion and in constant circular motion
Let the motion of the particle be in the direction of the o^-axis, then
we have / 2 / 3 = 0, i.e.
Vt - (A, 0, 0,/ 4 ), U t = (A, 0, 0,/ 4 ). (145)
Further, we shall assume that the particle has zero velocity in S at the
time r and that S'(r) coincides with S at that time. On account of
the equation U l U l = U\+U\ = c 2 we may therefore write U l in the
form U l - (csinh0(r), 0, 0, iccosh0(r)), (146)
where 9(r) is an arbitrary function of r which is zero for r = 0. Hence,
/ r r \
/ t (r) = \c J sinh dr, 0, 0, ic J cosh 6 drl (147)
^o o '
Further, if 4^( T ) represent unit vectors in the directions of the spatial
axes of the successive rest systems S'(r) we have
40(0) = g fci for r = 0. (148)
Each of these vectors satisfies the equations (144) ;
/JpW
therefore ^- = (epUg)U k /c*. (149)
dr
IV, 47
122 FOUR-DIMENSIONAL FORMULATION OF
On account of (145) we see at once that
are solutions of (149). To find the components e ( 1 1) and e^ 1) we use the cir-
cumstance that
_ Qj e o )e u) = l (151)
are integrals of the equations (149), as is seen at once by multiplying
(149) by U k and e^ respectively. Hence
= 0,
= -c
e u>2,
(152)
i.e.
1C
(153)
(154)
From (124) and (124') we therefore simply get
'UJic iUJ^
0100
0010
'JJic UJic/
which correspond to special Lorentz transformations (see (16)).
By means of (154),. (146), and (147) the transformation equations
(140) may be written
x l = c si
Xn
Xo
(155)
In the special case where the motion of the particle is hyperbolic
we get from (III. 46)
T
# 4 = ic I cosh dr-\-x\ i sinh 0(r) -\-x cosh 0(r)
i.e.
t =
(156)
IV, 47 THE THEORY OF RELATIVITY
Thus, from (III. 47) and (156)
123
= 0, / 3 =
o
= i sinh~-
9 c
\ c
1 \ c c
A comparison with (146) shows that d(r) in this particular case is
and the transformation (155) reduces to
(157)
(158)
(159)
*-' i ' *
x, cosh- .
g\ ^
i /
tX/2 v *'2> 3 3
x i-- sinh -- -4- x\ i sinh - 4- x* cosh --
(160)
9 c ' ' c
By means of the transformation coefficients (154) we get for the com-
ponents U[ of the four acceleration vectors of the particle in the succes-
sive rest system $'(T)
/: - |A . u k - l~* fll +~u u*, o, o,
?A * \ ic c
or, on account of (158),
#; = (?, 0,0,0), (161)
which shows that the accelerations of the particle in the successive rest
systems are constantly equal to g (see equation (42 ; )).
We shall now briefly discuss the solution of the equations (136) in the
case where the particle is moving in the (x l x 2 ) -plane with constant
angular velocity aj in a circle of radius a. In this case we have
withy = (lu*/c 2 )-*,u = acu being the constant velocity of the particle
in the circular motion. From (162) we get
J sin(coyr), 0, <
124
FOUR-DIMENSIONAL FORMULATION OF
IV, 47
By a straightforward calculation it is easily verified that the following
scheme of coefficients a lk provides a solution of the equations (136):
cos a: cos
sin a: sin
1AV
sin a cos j8 y cos (\suijS i^-
coy a sin/? y sin a cos/? sin asin/J-fy cos a cos/?
01
t - sin a
c
uy
i - cos a
c
c
!^ c
c
7
(164)
with at a>yr, j8 ya a>y 2 r.
The transformation AS Y -> S'(r) is then obtained from (139) or (140) by
introducing the expressions (164) for oc lk .
For r = we get
(165)
Thus, denoting the space-time coordinates of the system <S"(0) by x , we
get from (139)
^i == x "' ^2 = y x \i #2 I
(166)
^3 "^ ^3? 3*4 ==* 0: 2 + y^4 I
C /
which represents a special Lorentz transformation from S to a system
S'(0) moving with the velocity u in the direction of the # 2 -axis.
At the later time r -= r t -= 27r/coy we get from (164)
1
r
ay.
c,
1
.u
r
cos
ysin^ i
uy
I
\ _ l. COS /
c
-0
r
(167)
where/?! 27r(y 1). The coefficients a ^(T^) may, however, be written as
IV, 47 THE THEORY OF RELATIVITY 125
(cos ft sin ft 0\
sinft cosft (16g)
I/
If the space-time coordinates of the system S'(T}) are denoted by x\ we
or x\ = cosft #5- sin ft x,
#| sinft zj+cosft a?g, (169)
Thus the system S'^^) does not coincide with the system S'(0), but has
to be rotated through an angle ft in the (a^a^) -plane in the direction of
motion of the particle in order to give the spatial axes of ^'(TJ) the same
orientation as the axes of $'(0) or as the axes of S. In other words, the
axes of S have to be rotated through an angle ft = 2n(y 1) in the
(^ 1 x 2 )-plane in order to give them the same orientation as the axis of the
system $(TJ). This is due to the Thomas effect. Integrating the formula
(II. 65) for the velocity of the Thomas precession over a whole period
T, we get T T
f f VX V
o> dt (y 1) ~dt
J J v*
and, since v x v in our case is a constant vector perpendicular to v and v
and vT = vwT 27rv, the total angle of precession is just
9 = -27r(y-l) = -ft. (170)
48. Tensor and pseudo- tensor fields. Tensor analysis
As in ordinary space we speak of a tensor field of rank n in (3+ 1) -space
if to any point in this space is connected a tensor of rank n. In particular,
we have a tensor field of rank zero, a so-called scalar field, if an invariant
number is connected with every event point. This means that we have a
certain function of the coordinates <f>(x) <(# t -) = ^(x^x^x^x^) in
every system of coordinates S, such that
<f>'(x') = <f>(x), (171)
if <f>'(x') is the function corresponding to the system of coordinates S'
and the numbers (x[) and (X T ) are coordinates of the same event point in
the two systems S' and S, respectively. In general, <f>' will be a different
function of the variables (x() than <f> is of (x^. Thus a scalar function
^( x i) i s generally not a form -in variant function of the coordinates. This
126 FOUR-DIMENSIONAL FORMULATION OF IV, 48
will in fact only be the case if <f> is a function merely of the quantity (4)
which is also invariant in form.
Analogously, we have a tensor field of rank 1 when a four- vector is
connected with every event point. The components a t (x) and a((x') of
the four-vector in two arbitrary systems of coordinates 8 and S' will
then be functions of the coordinates of the event points, and
a\(x') - * lk a k (x) (172)
when the connexion between the variables (x 1 ) = (x[) and (x) (x^ is
given by (3) and (13).
For tensor fields of higher rank equations exactly analogous to ( 1 7 1 )
and (172) are valid.
Now from an arbitrary scalar field <f>(x) we can by means of a co variant
operation form a vector field with the components d(f>jdx l in the arbitrary
system of coordinates S\ for, from (171), we obtain by differentiation
d ^_ ^ <tf 3x k ^ a ty (173 )
~dx[ r dx k dx\ lk dx k
dx
where we have used - oL lfe , (174)
following from (13).
The four-vector d<f>/dx l is called the gradient of <f> and is written
grad^ = g-. (175)
It is analogous to the ordinary gradient vector in three dimensions.
Similarly, from a vector field a t (x) we can form a tensor field of rank
2 with the components da i jdx k in an arbitrary system of coordinates,
for by differentiation of (172) we get
8x m 8a l da t
(176)
The antisymmetrical combination da k /8x l da l /dx k is also a tensor field
of rank 2 which is called the curl of the vector field a^x). It is denoted by
curl^a = curl tjfc {aj = ^-^- ( 177 )
(In the expression curl lA .{aJ we shall of course not sum over i.) The curl
is an antisymmetrical tensor. Therefore, in three-dimensional space it
corresponds to an axial vector, viz. to curia.
IV, 48 THE THEORY OF RELATIVITY 127
By contraction of the tensor field dajdx k we obtain a tensor of rank
zero. From a vector field a^x) we can thus construct a scalar field
^ *$, (178)
8x1 Sx(
which is called the divergence of the vector field a { . It is denoted by
} = (179)
and is analogous to the ordinary three-dimensional divergence div a.
If a l is the gradient of a scalar /f, i.e.
^A MQH\
a l = - , (180)
the divergence of a l becomes
Cd* G UJ . , . / 1 f\ i \
_* r_ = QA (181)
f dx i 3x T dx l
where we have put
I I __ _ . __ ^ m (182)
The operator (182), d'Alembert's operator, is thus a covariant operator;
it is the four-dimensional generalization of Laplace's operator
A = ai'
In the same way, by means of differentiations we can always form a
tensor field of rank n+ 1 from a tensor field of rank w, and by subsequent
contraction we then get a tensor field of rank nl. As in the special
cases just considered, this is a consequence of the transformation equa-
tions for tensors together with the equation (173) and the orthogonality
relations (11) and (14). From a tensor field t ik of rank 2 we can thus
construct a tensor field dt ik /dx l of rank 3, and by contraction we obtain
a tensor field of rank 1, i.e. a vector field
div 7 . = div.ft-J = ~, , (183)
ui*/ dXk
which is called the divergence of the tensor field t ik .
If the tensor field F lk is antisymmetrical, we can obviously form a
completely antisymmetrical tensor field of rank 3
ourl, w * = curU**} = + + > < 184 >
128 FOUR-DIMENSIONAL FORMULATION OF IV, 48
which is called the curl of the tensor F lk . If we interchange any two of
the three independent indices in (184) the expression changes sign.
Thus, curl lkl F is zero if two of the indices are equal and the tensor
41
cur\ ik i has only =4 independent components. If F llc is equal to
the curl of a vector field, i.e. if
n 1 d a ic & a i /ior\
^ = eurl l4 =_*-^i, (185)
the curl of F ik is identically zero, i.e.
curl lW JF = 0. (186)
In the same way we speak of a pseudo-tensor field when to each point
in space is connected a pseudo-tensor. These are also frequently called
tensor densities, since t lk dx l dx 2 dx^dx^ according to (121), is a tensor
when t lk is a pseudo-tensor. Thus a pseudo-scalar is a tensor density of
rank zero, a pseudo- vector a tensor density of rank 1, etc. To the various
antisymmetrical tensor fields can be attributed dual tensor densities
which have the same reciprocal connexion as have the tensors and their
dual pseudo-tensors defined in 44. The vector density dual to curl, A ^
is equal to div t F*, where F* k is the tensor density dual to F ik , for,
according to (108), we have
'dx ~~ 2i lklm fa ~^ iV lk
UJL> r. ** (/ \JJL' i. lnj
which just means that div, F* and curl lfr/ F are dual to each other
49. Gauss's theorem in four -dimensional space
If a a(x) is a three-vector field and V a domain in 3-space bounded by
a closed surface a, Gauss's theorem in ordinary space is expressed by
the equation
[divarfF I a n da, (188)
F a
where a n is the component of a in the direction of the outward normal
n to the surface element da. (An elementary proof of this theorem is
given in Appendix 1). Gauss's theorem thus permits of transforming
the volume integral o'n the left-hand side of (188) into an integral over
the two-dimensional boundary a of the volume V. If n is a unit vector
in the direction of the outward normal, (188) may also bo written
x (189)
V
where dV is given by (107).
IV, 49 THE THEORY OF RELATIVITY 129
If we choose the surface element in the form of a parallelogram formed
by infinitesimal vectors dx L and 8x t lying in the surface a, the surface
element may be represented by the tensor d<r lK = dx t &jc K - rfx K S.r t
obtained from (99) by substituting dx L and 8x L for a L and 6 t , respectively.
Alternatively the infinitesimal parallelogram may also be represented by
the corresponding axial vector do l defined by (100). Since this vector is
perpendicular to the surface element, ( 1 89) can also be written in the form
J ' dv = / a ' d i = J a ' 8 <
l
^ xx (I90)
provided that the sequence of the vectors dx t and Sx t is chosen so that the
axial vector da t lies in the direction of the normal pointing away from
the domain V.
In this form Gauss's theorem may be immediately generalized to four
dimensions.*)* If a^x) is a four-vector field and X a domain in (3 f-1)-
space bounded by the closed three-dimensional surface V, Gauss's
generalized theorem takes the form
J ds = J a * dVi ^\\ (ti
i
whererfS is given by (121) and (dx t ), (8x t ), and (Aa^-) are three four-\ ectors
lying in the boundary space V. The pseudo-four-vector dV t is given by
(114) with (a t ), (6 t ), (c t ) equal to (dx t ), (8^), (Aa; t ), respectively.
If a part of the boundary V is a hyperplaneQ defined by .r 4 -- constant,
the vectors (dx t ), (8^ t ), (Ao: t ) are orthogonal to the time axis and we can
Ch se cte t = (fo lf 0,0,0),
8a? i = (0,rfe 2 ,0,0),
A^- (0,0,^,0).
The pseudo-vector (W l on the hyperplane O will then have the components
dV t (0,0,0, idx l dx 2 dx^) 9 (191')
where the plus or the minus sign should be taken according as the
normal to 12, pointing away from the region S, lies in the direction of
the negative or the positive time axis.
If t lK is a space tensor, we have by analogy with ( 1 89) and (190)
dv = J /w n * da ^ / '* da * = J iat 8 ^ ^ A 8 ^' ( 1 92)
a a a
f A. Sommerfeld, Ann. d. Pfnjs. 32, 749 (1910); 33, 649 (1910).
3595.60
130 FOUR-DIMEXSIOXAL FORMULATION OF IV, 49
and the analogous equation in (3 |-l)-space is
f ^rfS = (t lk dV k ^ [i iK Z klmn d^x m ^ n . (193)
J <>x k J J
50. The fundamental equations of mechanics for incoherent
matter
As a first application of the mathematical methods developed in the
preceding sections we shall now consider the motion of continuously
distributed matter under the influence of given external forces. In order
to be able to apply to such a system the fundamental equations of motion
of material particles, deduced in Chapter Elf, we shall regard a con-
tinuous mass distribution as a limiting case of a distribution of a very
large number of material particles. If the particles are so small and the
number of particles per unit volume is so large that our macroscopic
measuring instruments cannot distinguish between the single particles,
the mass distribution in an arbitrary system of inertia 8 can be described
by a mass density /^(x, /) which, for practical purposes, may be regarded
as a continuous function of the space and time variables, /^(x, /) is defined
so that /zSF is equal to the total mass inside the \olumo element SI 7
at the place x and at the time /. The motion of the matter at any place
and at any time is described by a velocity vector u ^- u(x, f), which is a
function of x and / Then the mass current density is equal to /zu. In
Newtonian mechanics the mass is a quantity which is conserved. Accord-
ing to formula (III. 22) this is not so in the theory of relativity, when the
matter is acted upon by forces, since the velocity of a given material
particle, and thus its relativistic mass, in this case changes with time. On
the other hand, the proper mass of the matter is conserved in many
cases, viz. when the four-force at any point is orthogonal to the four-
velocity of the matter at the place considered, i.e. when the equation (57)
holds everywhere and at any time.
The density of proper mass /x , i.e. the proper mass per unit volume, in
the arbitrary system of inertia 8 is, according to (III. 22), connected
With ^ Mo ^Vd- 2 /c 2 ). (194)
/x , like [A, is a given function of x and t.
Now, considering a certain point inside the matter at a given time, we
can introduce the system of inertia 8 relative to which the matter at
this point is momentarily at rest. In 8 (194) reduces to
/*8 = /*, (195)
where all quantities referring to /S Y are provided with the superscript 0.
IV, 50 THE THEORY OF RELATIVITY 131
In the rest system the proper mass density is thus identical with the
relativistic mass density. In contradistinction to ^ , juJJ = //, is thus an
invariant. Further, it is also easily seen that fJL Q ^J( 1 u 2 /c 2 ) is an invariant.
For, if we consider a small piece of matter which in 8 has a volume SF,
in the rest system 8 it has a volume SF which, according to (II. 34),
is connected with 3V by the equation
SF- 8VJ(I-u*/c*). (196)
The invariant proper mass of the material particle expressed in the two
systems is then given by
/i sr --= ^sro-^sp, (19?)
which, by means of (196), leads to the equation
^/c 2 ) - M =- invariant, (198)
or, according to (194), to /z l U - == /i. (199)
Consequently the quantities on the left-hand side of (198) and (199) must
be in valiants.
Let now </> </>(x, t) be a given function of the space and time coordi-
nates in A r . We must then distinguish between the local time differentia-
tion c>(f)/('f indicating the change of <f> per unit time at a fixed point in
space and the substantial differential coefficient d(f>/dt indicating the
change per unit time when we follow the matter in its motion. We
obviously have 7 , ,.,
(200)
In the same way we have for a three-dimensional vector field a a(x, t)
VV ., (201)
at ct
since an equation of the type (200) must hold for each component of a.
Let us now consider the matter which at the time t is situated inside a
closed surface a enclosing a domain with the volume V. At the time
t-\ dt the volume of this material will have been increased by an amount
dV = dt j u n da, (202)
(7
where the integral on the right-hand side is a surface integral over the
surface cr, and where u n denotes the component of u in the direction of
the outward normal to the surface element da', for every face element da
moves a distance of u dt during the time dt and therefore sweeps through
132 FOUR-DIMENSIONAL FORMULATION OF IV, 50
a volume u n dtda during this time. By means of Gauss's theorem (188)
equation (202) can be written
dV ^ J -' udV, (203)
an equation which must hold for every part of the material body. Con-
sidering in particular the matter which at the time t lies inside an in-
finitesimal volume element 8F, we get for the volume dilatation per unit
(204)
OV ill
Let us begin with the consideration of a system in which the proper
mass is conserved. For any material particle with the volume 87 at the
arbitrary time t we must then have
o o 0, (205)
which, by means of (204), can be written
% + Mo divu-0. (206)
at
This equation which expresses the conservation" of the proper mass can
also be given another form when we apply equation (200) to the function
/x (x,Z). We then obtain
0,
or + divOioU) = 0. (207)
ct
As /z u is the current density of the proper mass, (207) is the continuity
equation expressing that the proper mass in the system considered has
neither sources nor sinks.
By means of (39) we can now define a four-velocity U t at any point in
the matter and at any time. As U t i^(x) is a function of the space-
time coordinates, we thus have a four-vector field in (3-f-l)-space. In
the same way, the invariant mass density /u, defined by (195) and (199)
can be regarded as a scalar field in (3-f 1) -space, H Q being then a function
of the space-time coordinates in any system of inertia:
w 2 (ar)/c 2 } = p*(x). (208)
IV, 50 THE THEORY OF RELATIVITY 133
By multiplication of the scalar /x/c and the vector U T we obtain a new
four-vector if0 rr
c. = ^f, (209)
which may be called the four-current density of proper mass. According
to (39) and (198) we get for the components of c i
(210)
and the continuity equation (207) can be written in the tensor form
^i^i^ffi^O. (211)
dx t c ox i
The left-hand side of (21 1) is equal to the four-dimensional divergence
(178) of the four-current density, and the covariance of this equation
under rotations in (3+l)-space is therefore evident.
The forces acting on the different parts of continuously distributed
matter are partly external impressed forces, partly elastic forces acting
between neighbouring parts in the continuum. In this chapter, we shall
completely neglect the last-mentioned forces and postpone the con-
sideration of the elastic forces to Chapter VI. Here we therefore treat
the matter as a kind of incoherent dust. The impressed forces, however,
are assumed to be volume forces which in every system of inertia S can
be described by a force density f so defined that f 8 V is equal to the force
on the volume element 8F.
Let us now consider the motion of a small material particle with the
volume 8V arid with the proper mass JJL Q 8V = /LL SF. When U L is the
four-velocity, the four-momentum of this particle is, according to (50),
Pl = ^o 8FC7 t , (212)
and for the four-force (54) we obtain by means of (196)
*r*> -f7r^rx\ = ( f > ^^) 8F - < 213 )
& I si\ III ni I s*\ I I st I ^ '
Since 8F is an invariant and 2^ is a four-vector, the quantity
will also form a four- vector, the four-force density. The spatial com-
ponents of/ t are equal to the ordinary force density and/ 4 is equal to the
mechanical work done per unit of time and of volume multiplied by ijc.
134 FOUR-DIMENSIONAL FORMULATION OF IV, 50
The motion of the particle considered is now described by equation
(55). Substituting from (212), (213) in (55) gives, when the proper mass
is conserved, i.e. when - (p? SF) 0,
P-^=f l , (215)
where f t is given by (2 1 4). Since /x and dr are invariants, both sides of this
equation represent four-vectors. The first three equations (215) are
again the equations of motion, while the fourth equation expresses the
theorem of conservation of energy. According to (214) and (39)
which, as discussed in 38, is essential for the conservation of the proper
mass. By multiplying (215) by U 1/ and summing over i, the left-hand
side becomes zero on account of (41'), and the equations (215) there-
fore appear to be compatible with (216).
The fundamental equations of mechanics assume the simple form
(215) only in systems in which the proper mass is conserved. If we drop
this assumption we have, in accordance with (59), to use the following
expression for the four-force density:
(217)
where q is the amount of non -mechanical energy developed per unit of
volume and time, so that (f.u)-f- q represents the total effect per unit
volume. In this case the left-hand side of (55) becomes
. (218)
ar dr dr
Since 8F is the volume of the particle and dr is the time increase, both
measured in the rest system *S f , we obtain by applying equation (204) in
the rest system
= Srdivu , (219)
dr
where div means differentiations with respect to the space coordinates
(X Q L ) in /S. Tn the rest system u = 0, on the other hand, the derivatives
of u with respect to the coordinates need not be zero. By means of the
expresvsions (39) for U l and t/J we now get, since the quantity dUJdx l is
invariant, ~ n ~r 7 o
^ = ^ = divu. (220)
dx k 'dx\
IV, 50 THE THEORY OF RELATIVITY 135
Using the equations (217)-(220) in (55) we then obtain in the general
case, instead of (215),
=L (221)
where / t is now given by (217). Since, furthermore,
i'x^ dr
(221) can also be written
JL^of/.^) _._f^ (223)
which thus represent the fundamental equations governing the motion
of a continuous mass distribution in the general case in which the proper
mass may not be conserved.
From (217) and (39) we now obtain
where q is the non-mechanical effect in the rest system. This quantity
of course is an invariant, and we ^et from (224)
q-M(l-u*lc*) (225)
Multiplying this equation by the invariant
8 TAJ - SFAr
we again have the equation (66), since &Q ~ q SFAf is the amount of
heat conveyed to a certain material particle during the time A. If we
multiply (223) by U^ and sum, we obtain, using (41), (41'), and (224),
or, according to (209) and (210),
These equations are thus generalizations of the equations (211) and (207)
for the case where the system contains sources for the proper mass, and
the expression <//c 2 tor the source density is in accordance with Einstein 's
general theorem (III. 74).
In the preceding considerations it has been tacitly assumed that no
heat conduction takes place in the matter, so that the transport of heat
therefore occurs only by means of convection. Fn the more general case
where heat conduction also occurs, it must be taken into account that
136 FOUR-DIMENSIONAL FORMULATION OF IV, 50
the energy transported by means of heat conduction represents an extra,
non-material momentum which would cause a change in the first three
equations (223).
51. The kinetic energy-momentum tensor
The quantity appearing on the left-hand side of (223)
e ik = vU t V k (228)
is a symmetrical tensor of rank 2 which is called the kinetic energy-
momentum tensor. f (223) can thus be written
f^ = / (229)
cx k
i e. the four-force density is equal to the divergence of the kinetic energy-
momentum tensor. By means of (39), (198), and (199) we obtain the
following expressions for the components of this tensor:
= -^ = -h, (230)
where h is the energy density or, more precisely, the sum of the kinetic
energy and the proper energy per unit volume. Multiplication of (230)
by 8V makes the right-hand side, apart from the minus sign, equal to
the energy of the small material particle inside the volume 8V.
The three components 6 4l form the components of a spatial vector
which can be written
(231)
The three components 0, 4 can in the same way be written
(^^^) = -^^ r -g. (232)
hi view of the equations (229) (cf. (236) and (238)) the quantities
**
,0 4l = 7m and OJic = g
must be interj)reted as energy current density and momentum density,
i espectively. The symmetry of the energy-momentum tensor which
involves t4 = 4t or 7
g = - 2 u (233)
c
then simply expresses that the energy h corresponds to a mass h/c 2 .
f H. Mmkowski, GotL Nachr., p. 53 (1908); Math. Ann. 68, 472 (1910).
IV, 51 THE THEORY OF RELATIVITY 137
The spatial part 6 iK of the energy-momentum tensor can be written
Q p, G U L U K __ f9<U\
"<* - jir^ict) ~ g ^ (234)
where g L and U K are the components of the space vectors g and u.
Just as the quantities (231) represent the energy current density, each
row of the space tensor IK in (234), e.g.
(0*AM = giV> (23- r >)
can be interpreted as the current density of the momentum component y t .
Therefore IK is also called the momentum current tensor.
The fourth equation (229), expressing the energy conservation, can
now, by means of (230), (231), and (217), be written
+clivOzu) = . (237)
This equation is exactly analogous to equation (227). While q/c* ex-
presses the source density of proper mass, we see that the source density
for relativistic mass is ((f .u)+^)/c 2 . Again this is in accordance with
Einstein's relation (III. 74).
In the same way, the first three equations (229), the equations of
motion, can be written
%4.^~ f (238)
fl+c^-'*'
^ + div(i/ 4 u)=/ i , (239)
where we have made use of (217), (232), and (234). Equation (239) plays
the same role for the momentum as does the analogous equation (23(>) for
the energy. It regulates the flow of momentum in the material con-
tinuum, the force density f appearing here as a source of momentum.
If we multiply (238) by X K and subtract the corresponding equation
obtained by interchanging i and /c, we get
r\ r\
^ t (9^ K -g K x i ) + -^~(0 iX x K -0 KX x l )-0 lK +0 KI> - f L x K ~f K x L . (240)
On account of the symmetry of the tensor d LK , the last two terms on the
left-hand side cancel each other. Introducing the angular momentum
density and the density of momentum of force by
>IK = ^if/K-^f/p (241)
138 FOUR-DIMENSIONAL FORMULATION IV, 51
(cf. equations (82)-(86)), (240) can be written
? w+?*^ = d > (243)
where we have applied the expression (234) for LK . This equation can
also be given the form
^+(u.grad)m w +m l(t divu - d iK , (244)
and, when we multiply (244) by BV, we obtain, by means of (200) and
(204), ,
* t (n, lK W)^d lK ?>V. (245)
This equation expresses the angular momentum theorem for a small
material particle with the volume 8V. We thus have seen that the
symmetry of the kinetic energy-momentum tensor is a very important
property, since the symmetry of the spatial part is essential for the
validity of the angular momentum theorem in its usual form, while the
equation t4 -~ 4t , i e. (233), is an expression for Einstein's theorem of
the inertia of energy.
The kinetic energy -momentum tensor satisfies the relation
o* u k - ^ u k u k =. -pw\ = , _ w /t) (246)
where h Q is the energy density in the rest system.
ELECTRODYNAMICS IN THE VACUUM
52. The fundamental equations Of electrodynamics in the
vacuum. Four -current density for electric charge
IN Chapter III we have seen that it is necessary, to change the funda-
mental equations of mechanics in order to bring them into accordance
with the principle of relativity. This is not so with the equations of
electrodynamics in the vacuum, the Maxwell equations, which, as we
shall see, are already co variant under Lorentz transformations. f
Let us imagine two teams of experimental physicists who have in-
stalled their laboratories in two different inertial systems S and 8' and
who independently are performing electromagnetic experiments. By
means of electrically charged test bodies and magnetic compass needles
the physicists in S will be able in the customary way to determine the
electric field vector E and the magnetic field vector H as a function of
the space-time coordinates x and t in S. By the same procedure the
physicists in 8' will be able to determine electric and magnetic field
vectors E' and H' as functions of the coordinates x' and V in S'. Further-
more, the two groups of physicists can, independently of each other,
determine the charge densities p and p in S and S' . In the present
chapter we shall consider only electromagnetic phenomena in the vacuum ;
as neither conductors nor dielectric and magnetic substances are present
here, the only type of electric currents occurring are convection currents.
The current densities in 8 and 8' will thus be pu and p'u', where u and
u' are the velocities with which the charges move in 8 and S', respec-
tively. All these quantities will be certain functions of the space and
time coordinates in 8 and 8'.
Now, according to the principle of relativity, the equations deter-
mining the fields as functions of the charge distribution should have the
same form in 8 and 8' . ( Consequently, both groups of physicists should as
a result of their experiments be led to the Maxweli-Lorentz field equa-
tions for empty space. In S we thus have, applying Heaviside's units,
1 ,^TT
-~^0, (la)
c dt
i^ = ^, (16)
c dt c
t H. Fomraip, d.It. 140, 1504 (1905), Rend. Pal. 21, 129 (1906), A. Einstein, Ann. d.
Fhys. 17, 891 (1905), H. Mmkowski, see ref., Chap. IV, p. 136.
140 ELECTRODYNAMICS IN VACUUM V, 52
and in S' we get equations resulting from (1) by adding a prime to all
quantities. The equations (1) are identical with the fundamental equa-
tions of Lorentz's classical electron theory.
While the connexion between u and u' is given by (II. 55), we do not
yet know the connexion between p and />' or between E and H, on the
one hand, and E' and H' on the other. However, it is one of the most
fundamental experiences that electric charge is conserved, a property
which, in analogy to (IV. 207), can be expressed by the continuity
equation ~
( P u) = 0. (2)
This equation is a simple consequence of Maxwell's equations (J6).
Obviously, a similar equation must be valid in S f , viz.
? + divVii') = 0. (2')
The connexion between p and p must now be such that for arbitrary
charge and current distributions (2') is a consequence of (2). Defining
four quantities in S by
(3)
and analogous quantities s[ in S', (2) and (2') may be written
= (4)
and ~ = 0. (4')
In Appendix 2 it is shown that, if (4') is to be a consequence of (4) for
all possible charge and current distributions, the connexion between
s t and s[ must be given by
where oc lk are the coefficients in the coordinate transformation (IV. 3)
connecting the systems S and S'. Thus, s i is a four-vector, called the
four-current density, and (4) expresses that the divergence of the four-
current density is zero (cf. IV. 178).
/ If we multiply the invariant
s i s l = $Xi (6)
by 1, we obtain with the help of (3) an invariant
- P 02 , (?)
V, 52 ELECTRODYNAMICS IN VACUUM 141
where p is the charge density in the rest system $. Hence we get
By means of (8), the equation (3) can be written in the form
,=^,
where U l is the four-velocity defined by (IV. 39). Equation (9) is quite
analogous to the expression (IV. 209) for the four-current density of the
proper mass.
Let us now consider the charge p 8V connected with a material volume
element 8V. If SF denotes the corresponding volume in the rest system,
we have 8F SF\/(1 u 2 /c 2 ), which together with (8) gives
pSF-/>3F. (10)
Hence, the electric charge of a certain material volume element is an
invariant, and the same is therefore also true for the total charge of a
material body. This important theorem of the invariance of electric
charge is thus a consequence of the validity of the continuity equation
in every system of inertia. It can also be made plausible by the following
reasoning. Consider a charged particle of charge e originally at rest in S.
Under the action of a force the particle is accelerated until it has the same
velocity v as has S' relative to S. Since the charge of a particle is con-
served during acceleration, the particle has still the charge e relative
to S. On the other hand, the particle now has the velocity zero
relative to S' and, since it is now in the same situation relative to S f
as it previously was relative to S, the charge e' of the particle rela-
tive to S' must be assumed to be equal to the constant charge e relative
to S. Therefore we must have at any time e' = e, in agreement with
the equation (10).
53. Covariance of the fundamental equations of electrodynamics
under Lorentz transformations. The electromagnetic field
tensor
In every system of inertia S we now define a quantity F lk by the
equations
F ik = -F kl , (F m , F n , F 12 ) = H, (F n , F , F n ) = E, (11)
i.e.
H. -H u -iE x
H v -H x
* Tji /* f
l&y 1>& Z
142 ELECTRODYNAMICS IN VACUUM V, 53
The equations (1 a) can then be written
CXt O^r O^k,
Since the expression on the left-hand side in (13) is completely anti-
symmetric in the three indices i, k, /, (13) represents only four indepen-
dent equations which are obtained, for example, by putting (i, k, I) equal
to (1,2,3), (4,2,3), (4,3,1), (4,1,2), respectively. It is easily verified
that these four equations are identical with the four equations (I a).
From the general validity of the equations (4) in every system of
inertia we concluded that the quantities s t are the components of a four-
vector. In the same way we can conclude that the quantities F tk must
transform as the components of an antisymmetrical tensor if the equa-
tions (13), as should be required, are valid in every system of inertia.
The tensor F lk thus defined is called the electromagnetic field tensor, and
the equations (13) or (la) express that the curl of this tensor is zero
(cf. IV. 184).
On account of (IV. 187) the equations (13) may also be written
r v* ^ n
div t JP* = - '* = 0, (14)
where F* L is the pseudo-tensor dual to F lk obtained by the substitution
E -> H, H -* E in the expression (12) for F lk .
The connexion (11) between F lk and E ? H is the same as in (IV. 80, 80' ),
which indicates that E and H behave as a polar and an axial vector,
respectively, under pure spatial transformations. For a general Lorentz
transformation without rotation we have the transformation equations
(IV. 81') for E and H. These equations may also be written in the form
H' =
(15)
The division of the field into an electric and a magnetic field, which is
forced upon us by our measuring instruments, thus has no absolute
meaning. If, for example, we have a purely electrostatic field in S, i.e.
H = 0, there will, according to (15), be a magnetic field H' =/- in S'.
V, 53 ELECTRODYNAMICS IN VACUUM 143
This is also quite clear from a physical point of view, since a purely
electrostatic field in S' means that all charges are at rest relative to S.
Relative to S' the charges will therefore move with the velocity v.
Consequently we have in 8' a stationary current which causes a magnetic
field in S'.
By means of (3) and (12) the second set of Maxwell's equations (1 6)
can now be written ~ v
Since the left-hand side in (16) is the divergence of the electromagnetic
field tensor (cf. IV. 183), the left-hand and the right-hand sides are- trans-
formed in the same way, viz. as a four- vector. Hence, the co variance
of the equations (1 b) is a consequence of the co variance of equations (la)
and of the continuity equation (4). This is a strong argument in favour of
the exact validity of (16) and it is seen, in particular, that the term
1 dE
~c~dt 9
Maxwell's displacement current, is absolutely necessary for the co-
variance of the equations (1ft).
If we form the di\ ergence of the vector equation ( 1 6) we get, on account
of the antisymmetry of the electromagnetic field tensor,
-^0, (17)
'dx l <dx k
i e. the continuity equation (4).
54. The four -potential. Gauge transformation
As is well known, E and H can, as a consequence of the equations (1 a),
be written in the form
H- curl A, E = grad<A i , (18)
c tit
where the vector potential A and the scalar potential </> can always be
chosen in such a way that they satisfy the Lorentz condition
divA + ~^-:0. (19)
C ut
This can be accomplished in every system of inertia. If we now define
four quantities A l in every system of inertia by
A-MA,^), (20)
144 ELECTRODYNAMICS IN VACUUM V, 54
(18) and (19) can be written
' = 0. (22)
dx l
Since F lk is a tensor, it must be possible to choose the potentials in the
different systems of inertia such that the A i are transformed as the
components of a four-vector, the four-potential. According to (21) and
(22) the electromagnetic field tensor is equal to the curl of the four-
potential which has a divergence equal to zero. Moreover, from (IV. 184-
186) it follows that Maxwell's equations (13) are a consequence of (21).
When F lk is given, the four-potential A L is by no means uniquely deter-
mined by (21); for, if A t satisfies (21), the functions
*-*'+%; {23)
where is an arbitrary scalar, will also satisfy (21).
The transformation (23) is called a gauge transformation and the
measurable quantities F lk are invariant under such transformations.
The Lorentz condition restricts the class of permitted gauge transforma-
tions, but there is still a great variety of potentials A l satisfying this
condition. Substitution from (23) in (22) leads to the condition
= D<A = 0. (24)
r '
If is an arbitrary solution of (24), A* will thus satisfy both (21) and (22)
if A l satisfies these equations.
Inserting (21) in (16) we get, using (22),
=*, (25)
dx k fa k l ' l ;
or D^t = ~ s t . (26)
Any solution of (26) which simultaneously satisfies (22) gives, by means
of (21), a solution of Maxwell's equations (13) and (16).
55. Four-dimensional integral representation of the four-
potential
The equations (25) have the form of usual potential equations in
(3-}-l)-space; the solutions can therefore be found by a method exactly
analogous to that applied in three dimensions. f We shall first write
f A, Sommerfeld, Ann. d. Phys. 33, 649 (1910).
V, 55 ELECTRODYNAMICS IN VACUUM 145
down the solution of the equations (25) on the assumption that all four
coordinates x t are real, so that the four-dimensional space is Euclidean.
Let *, = *.-*.(/) (27)
be a four-vector connecting a fixed point P with coordinates x^P) and a
variable point with coordinates x % . If R 2 = (R l R ( ) denotes the
square of the distance between these points a simple calculation shows
that , 02
at any point x l ^=- x z (P). Further, if *jj(x) and <f>(x) are two arbitrary
regular functions of (x) = (#J, we have
^QA-0n< = d d x
Now let us put t/r = 1/jR 2 in this equation and integrate over the whole
four-dimensional space which lies outside a (three-dimensional) sphere
R% = R-R (x x (P})(x x (P)) = cfi (29)
with radius a and with the point P as centre. If in the right-hand side we
use R i = x^x^P) as integration variables instead of # t , we thus get,
on account of (28),
1
x l dx 2 dx 3
(30)
where the integration has to be extended over the four-dimensional region
for which Rt^K^^a*. (31)
Since the integrand on the right-hand side of (30) is a sum of partial
differential coefficients, the integral can be transformed into a three-
dimensional integral over the sphere (29), provided that the function </>
vanishes sufficiently rapidly at infinity. For the first term on the right-
hand side of (30) we then obtain
-
(32)
where the integration is to be extended over the sphere (29). Accord-
ingly, ( )+ and ( )- means that R in the function in the brackets is to
be put equal to the values
3595.60 L
146 ELECTRODYNAMICS IN VACUUM V, 55
and the domain of integration over the variables R 2 , 72 3 , # 4 is defined by
the inequality ^ ^ , ^ ^ < fl2 _ (34)
We get expressions similar to (32) for the three other terms corresponding
to k = 2, 3, 4 on the right-hand side of (30); they are obtained from (32)
by cyclic permutation of the indices (1,2, 3, 4).
(The transformations just performed in (30) correspond to Green's
theorem in three dimensions )
Subsequently, letting a -^ 0, the volume of the three-dimensional
domain of integration in (32) \\ill tend to zero as a 3 and, since
1 1
R- " a 2 "
in the brackets ( )*, the second term inside the brackets will tend to zero.
Since, moreover, ~ , r>
-- " k (35}
^Z A ~ 5* -"**"' I'' '
we obtain for (32) in the limits ot very small , using (33),
!zPi) dR 2 dR. A dlt 4 , (36)
where (f)(P) is the value of the function <j> at the point P. The domain of
integration (34) is the interior of the sphere p\ ~-^ a' 1 . The three other
terms obtained by cyclic permutation of the indices 1 , 2, 3, 4 are obviously
equal to the first term. Therefore, introducing polar coordinates in the
integration, we finally obtain for the right-hand side of (30)
(37)
(38)
a
477 r ,, 2 . 2 7
~T v( a ~~~Pi)Pi "Pi "
o
and (30) becomes 47r 2 </(P) = f -L
where d*x is an abbreviation of dx l dx 2 dx^dx^. This equation holds for
any regular function </>. If in particular </> is the function A l satisfying
(26), we thus get the formula
" 5-,d 4 *, (39)
whicli allows us to calculate A l at an arbitrary point P in the four-
dimensional space when s t is known at every point.
Until now we have assumed the variables .r t to be real. However,
in actual physical problems the four-current density s t is not given for
real x 4 , but only for purely imaginary .r 4 values corresponding to t < t(P).
V, 55
ELECTRODYNAMICS IN VACUUM
147
In the complex # 4 -plane s t is thus given only in the fat-faced part of the
imaginary axis in Fig. 13. We therefore deform the original path of
integration along the real axis into a loop L around this part of the
imaginary axis, using the analytic continuation of the function s t in the
integrand of (39). The expression (39) will then still be a solution of (26).
4 - plane
ict(P)
-100
FIG. 13.
a i a i
(40)
(41)
Since
we get for the divergence of A t (P) in (39)
4772 ~df(p) = ~ J Sl ' a^T" d * x = J fal 5*
by partial integration and application of (4). The solution (39) thus
also satisfies the Lorentz condition (22).
From (21), (39), (40), and (35) we thus obtain for the electromagnetic
field tensor ^ P r,
2ntf u (P) = J < ^ * Si d*X. (42)
56. Retarded potentials. Lienard-Wiechert's potentials for point
charges
In performing the integrations in (39) the sequence of the integrations
may be chosen arbitrarily and we shall first perform the integration
over the ^-variable along the path L in Fig. 13, keeping (x v % 2 , x 3 )
148 ELECTRODYNAMICS IN VACUUM V, 56
constant. If r = |x x(P)j denotes the spatial distance between the
space points x and x(P) corresponding to the event points (x t ) and
we have
= (Xt-Xt(P)+ir)(Xt-Xt(P)-ir). (43)
Therefore the quantity 1/B 2 as a function of # 4 has a pole inside the
loop L, viz. the point = x ^_ ir ^
in the complex # 4 -plane. Since the integrand in (39) has no other poles
inside L, the path of integration can be deformed into a contour around
the point (44) and, by means of Cauchy's theorem, we then obtain
J ** x \ x t(*) W xi-=x t (P)-ir r
L
Consequently the potentials (39) assume the form
-^ dV, (46)
where the integration has to be extended over the usual three-dimen-
sional space. The function s l in the integrand is not to be taken at the time
t(P)> but at the time t(P)r/c, corresponding to the fact that all electro-
magnetic disturbances are propagated with the finite velocity c. There-
fore the potentials (46) are called retarded potentials. If in (39) and (45)
we had integrated along a curve obtained from L by reflection with
respect to the point # 4 = ict(P), we would have obtained another solution
of (26) corresponding to advanced potentials. This solution, which con-
nects the field at a certain point at a certain time with the future charge
and current distributions, has, however, usually no immediate physical
application.
Let us now consider the potential of a point charge e in arbitrary
motion, the coordinates V t of the point charge being given as functions
of the time *: x t = x t (<). (47)
The four potentials can then be obtained from (46), but, since s l in (46)
on account of the retarded time variable has a rather complicated
dependence on the variables of integration, it is easier in this case to go
back to equation (39). Here we can now first perform the integration
over the space coordinates and, since s t is zero everywhere but at the
points satisfying (47), we have, according to (3) and (IV. 39),
Si dx dx dx - l ( eU
* ~ l
f
J
V, 56 ELECTRODYNAMICS IN VACUUM 149
Here u = (dxjdt) is the velocity of the point charge given by (47), U i is
the corresponding four-velocity, and
All these quantities are now functions of t or of the purely imaginary
variable # 4 , but by analytical continuation they can also be defined
'outside the imaginary axis in the # 4 -plane. From (39) we thus get
L
The integrand again has a pole at the point (44) in the complex plane,
and near this # 4 -value the denominator has the form
R2 _d]P (x _.
~~ dx^ 4
where the functions
o p dHjf <y~P TJ 9 P TT V'* ^ / C ) /KA\
= K> k z = ^{Ifr Ujc ~T = ^*lk "k ' \*^^/
0/^4 a# 4 ao; 4 tc
and r(^) should be taken for the value of # 4 for which x^x^(P)-\-ir = 0.
By means of Cauchy's theorem we thus obtain|
Here (B k ) denotes the four- vector leading from the fixed event point P
to that point Q on the time track
* = *i(r) (52)
of the point charge in which the retrograde light cone
B 2 = B t B t = (53)
originating from P intersects the time track. Also U i should be taken at
the point Q.
- If r denotes the spatial part of the four-vector (JBJ leading from P to
Q we get for the denominator in (51)
77 R _ . t-i (u.r)+rc
1 '""VU-^/c 2 ) <J(l-u*/c*) ^/(i-u 2 /c 2 ) V ;
by means of (IV. 39) and (44). The equations (51) can thus be written
477A(P) =
eu/c
r+(u.r)/c
e
r+(u.r)lc
. (55)
f H. Mmkowski, see ref., Chap. IV, p. 136.
150 ELECTRODYNAMICS IN VACUUM V, 56
The equations (55) represent Lienard-Wiechert's potentials of a moving
point charge.
If we wish to calculate the electromagnetic field tensor F lk (P) at the
event point P by means of (51) we must keep in mind that the proper
time r corresponding to the point Q is a function of the coordinates of the
point P defined by the equation (53) or, according to (27) and (52), by
(xM-xAPMxM-xAP)) - 0. (56)
By differentiation of this equation with respect to x k (P) we obtain
Sr K k /K -v
(57)
From (21) and (51) we then get
toF (P) -*-!*-'*
( ' <h\U,R
or, using (IV. 41),
dU dl*
' u '-*^' (B8 >
Equation (58) could also be obtained directly from (42) if we first inte-
grate over the space coordinates and subsequently use Cauchy 's theorem
in the following integration along the curve L in the # 4 -plane. Here it
should be remembered, however, that the function l/7i 4 has a pole of
second order at the point (44).
Since F, k according to (58) has the form
^R^-E^, (59)
where a r is a four-vector, we have, according to (IV. 110, 111),
^^=0, (60)
where F* k is the pseudo-tensor dual to F lk , which is obtained from (12) by
the substitution (H, E) -> ( E, H). The equation (60) is thus identical
with the equation
(E.H) - 0. (61)
The electric and magnetic field vectors are thus everywhere and in
all systems of inertia perpendicular to each other. Further, we find by
V, 56 ELECTRODYNAMICS IN VACUUM 151
means of a simple calculation, applying (12), (58), and (53),
0/>2 r 4
( 62 )
Consequently the invariant |H 2 |E 2 | is always negative for the field
of an arbitrarily moving point charge. This, together with (61), involves
that for an arbitrary event point P we can always choose such a system of
inertia that the components of the magnetic field vector in this system
are zero at the point P. In order to obtain H' O'in (15) one needs only
to choose ,p TT\
v = r -p (63)
and this is physically possible, since
cEH ell
v = _ - = < c (64)
on account of (61) and (62).
57. The field of a uniformly moving point charge
Let us now in particular consider the field of a point charge moving
with constant velocity. The time track of the charge is here a straight
line in (3+ l)-space with a direction defined by the constant four-velocity
l^. Since dl\ldr --- 0, (58) reduces to
4nF. k = CC - (RJL R k U t ). (65)
(l^R^
By a suitable choice of the origin in the system of coordinates S we can
always ensure that the # 4 -axis and the time track are lying in the same
plane In Fig. 14, which gives a representation of this plane, the line L
represents the time track of the particle, and Q is the point at which the
retrograde light cone from the arbitrary point P intersects the time
track. Hence R ( is the vector leading from P to Q. If A is the projection
on L of the point P the vector leading from A to P, with components
^ l) , is orthogonal to the four-velocity l\, and x( l) is the projection of the
vector R l on a direction perpendicular to L\ hence
Therefore we have x ( *> R t -^ ( U t RJ. (67)
It is immediately seen that the equation (67) satisfies both equations
(66). From (67) and (53) we further obtain
152 ELECTRODYNAMICS IN VACUUM V, 57
In the rest system S' of the point charge the ^-axis is parallel to fy and
to the time track L. In this system we therefore have
^'-(r',0), (68)
where r 7 is the space vector connecting the point charge with the space
point />' corresponding to the event P. Since U t R l is an invariant we
get by means of (IV. 39), (27), and (44),
= cr' (r' = |r'|), (69)
FIG. 14.
in accordance with (67') and (68). Moreover, we get from (67)
R, U k -R k U t = -( x (
so that (65) can be written
(70)
In this expression we can obviously replace the vector x ( ^ by a vector
leading from an arbitrary point on the curve L to the point P, since all
such vectors have the form
x t = d*>+aU,, (71)
where a is a constant. In an arbitrary system of coordinates S we now
choose a vector x ( ^ such that the time component # ( 4 2) is zero. In Fig. 14
this vector is represented by the line BP. In $ we thus have
r,0), (72)
x\
,(2)
V, 57 ELECTRODYNAMICS IN VACUUM 153
where r is the space vector leading from the simultaneous position of the
point charge to the point of observation p corresponding to the event P.
(The r used here thus differs from the r used in the Li6nard-Wiechert
potentials which was the spatial part of the vector E^) We therefore
have
=-(U-lW), (73)
which by means of (12), (72), and (IV. 39) leads to-the following expres-
sions for the electric and magnetic field vectors:
er A tr e uxr tiA\
' = '
From Fig. 14 we see that the two vectors x( 2) and x( l} have the same
space components r' in S'. The connexion between r' and r is therefore
obtained simply by using the reciprocal of the transformation equation
(IV. 29) for the vector with components
Since the velocity of S' relative to 8 is u, we get
r' r I U ( U - r )(l V^""^ 2 / 02 )) ( nK\
I := I - j ^ / tJJ
hence r' = (r' + iHlE^U. (76)
Decomposing the vectors r and r' into components perpendicular and
parallel to u, respectively, (75) can be written
'* = /J -^' (77)
Hence the connexion between r and r' corresponds to a Lorentz con-
traction in the direction u (cf. II. 35).
The electric field vector lies in the direction of the radius vector r,
while H is perpendicular both to r and u. The equations (74) are easily
seen to be in agreement with the equations (61) and (62). The surfaces
with constant values of the quantity (E 2 ~H 2 ) are obviously rotation
ellipsoids, so-called Heaviside ellipsoids, which are obtained from
spheres r' constant by a Lorentz contraction in the direction of motion
of the point charge.
The expressions (74) may also be obtained in a much simpler way by
means of the transformation equations (IV. 81') for the electromagnetic
field vectors. If S' denotes the rest system of the point charge, the
154 ELECTRODYNAMICS IN VACUUM V, 57
electric field is spherically symmetrical and the magnetic field is zero in
S', i.e. ' er ,
4nE' = ~, H' = 0. (78)
Hence, from (IV. 81') and (78),
(UXr/)
Since the equations reciprocal to (75) are
(cf. IT. 35'), we again obtain the expression (74) for the electromagnetic
field of a point charge in uniform motion.
From the latter deduction it results that the equations (74) are valid
also everywhere outside a charged sphere provided that the charge
distribution in the rest system is spherically symmetrical En this case
r denotes the distance from the centre of the sphere while e is the total
charge on the sphere.
In the same way we can also determine the field of a 'uniformly
accelerated' point charge, i.e. a point charge performing a hyperbolic
motion (cf. 29). Again we can use two different methods- either we can
directly apply the equation (58) which is valid for an arbitrarily moving
charge, or we can introduce a system of coordinates 8* which follows the
charge in its motion and solve Maxwell's equations in this system, and
then transform back to the system S. However, the system of co-
ordinates $* will not be a system of inertia and, consequently, the
application of this latter method requires the development of a general
theory of relativity which allows the use of arbitrarily moving systems
of coordinates. The necessary tools for an application of the second
method will be provided in 97 and 115.
58. The electromagnetic forces acting on charged matter
By means of the equations deduced in the preceding sections we are
able to calculate the field created by an arbitrary charge and current
distribution. Now we shall consider the opposite problem of the influence
of a given field F lk on the motion of electrically charged matter. Our first
V, 58 ELECTRODYNAMICS IN VACUUM 155
task will be to determine the force on an electrically charged particle
of charge z moving in a given electromagnetic field with the velocity u
relative to a certain system of inertia S. In accordance with the method
outlined in 29 we shall now introduce the inertial system 8 in which
the particle is momentarily at rest. In this system the force F must be
F == eE, (79)
by the very definition of the electric field vector E in this system.
Introducing the equation (79) into (III. 43), and keeping in mind that
the velocity of >S' relative to S js equal to the particle velocity u, one
easily finds by means of the transformation equations (15) for the electro-
magnetic field vectors, Lorentz's expression for the force F in the system
S, viz. r ,
r , -,
= e E + -(uxH)L
(80)
This expression thus follows without extra hypotheses from the principle
of relativity.
This deduction becomes much simpler if we use the four-dimensional
representation and try to determine the expression for Minkowski's
four-force. In the case considered, the proper mass is conserved and the
four-force is thus defined by (IV. 54); therefore we have in the rest
system ^o ^ (F,0). (81)
If U t is the four-velocity of the particle we can now form a four-vector
with the components
*FikU k . (82)
In the rest system & we have
l/; = (0,0,0,fc), (83)
and the components of the four-vector (82) in 8 thus become, according
to (12), e
c ik k
From (79), (84), and (81) we see that the components of the four-vectors
F l and (82) are equal in the rest system, but two four-vectors whose
components are equal in one system of coordinates are altogether
identical. Therefore we must have
F t = -F lk U,. (85)
i c ik k \ )
in every system of coordinates.
156 ELECTRODYNAMICS IN VACUUM V, 58
Since F lk F ki we have
F t U t = e -F ilc UtU t = (86)
L>
in agreement with (IV. 57), and the equations of motion of the particle
are given by (IV. 56) and (85)
, ^ IK ^ K . v*'/
ar c
If we calculate the components of F l in (85) by means of (12) and
(IV. 39), we get the equations (IV. 54) with F given by the Lorentz
equation (80).
Now consider a continuous distribution of charged matter, with a
four-current density (3), (9), i.e.
\ _orr
(88)
in a given external field. The problem is then to find the expression for
the four-force density f l defined by (IV. 214). This question can be
solved on similar lines as above.
Consider a definite point in space at a definite time; the charged
matter at this point is moving with a certain velocity. Now, let 8 Q be
the momentary rest system of the matter at this point. The com-
ponents of s i in this system are then s = (0, 0, 0, ip) and the four-
vector
has the components F k s k = (pE, 0) (90)
in $. These components are equal to the corresponding components
(IV. 2-14) of the four-force in S Q
/? = (,0); (91)
for the force-density in the rest system must be given by
fo ^ p o E o (92)
by the definition of the electric field vector.
The four-force density must therefore be equal to the four- vector (89) in
every system of coordinates, hence
/* = ***>' (93)
From (IV. 214), (93), (88), and (12) we immediately get Lorentz's expres-
sion for the force density
(94)
V, 58 ELECTRODYNAMICS IN VACUUM 157
Further, we get, in view of the antisymmetry of the field tensor
/, U t = F tk s k U t = U t F lk U k = 0, (95)
c
which, as discussed in 50, means that the proper mass is conserved in
this case. Therefore the equations of motion have the simple form
(IV. 215). j TT
59. Variational principle of electrodynamics
Maxwell's field equations and the equations of motion (96) may be
derived from a certain variational principle formulated by Weyl and
Born.t If we put' F _ aA* 8A,
ik ~ dx t dx k '
where A i is the four-potential, the first set of Maxwell's equations (13) is
identically satisfied. Now, consider the invariant integral
L-JjSfdS (97)
12
with
02 l fi A k
L s , uc 2 = -
l l A-
(98)
integrated over a certain domain fi in (3+l)-space. s t is the four-cur-
rent density and /x is the invariant mass density in the rest system.
In the first place, we shall now consider an arbitrary variation 8A i of
the functions A^x) for which 8A l at the boundary of the domain
}, s i and JJL being kept constant by this variation. For the variation of
L we then get by partial integration, remembering that 8 - = - - ,
8L = J ^ ^ =, J (J
(99)
The condition SL = (100)
for any variation of the kind considered then leads to the equations
i.e. to the second set of Maxwell's equations (16).
Next we shall consider a variation in which the A l are kept constant
t H. Weyl, Raum-Zeit-Materie, Berlin, 1918. See also M. Born, Ann. d. Phys. 28, 571
(1909).
158 ELECTRODYNAMICS IN VACUUM V, 59
while the time tracks of the matter are varied. This means that only the
last two terms in (98) are affected by the variation. Let us fix our atten-
tion on a definite infinitesimal piece of matter with volume dV and
consider the infinitely thin tube of time tracks of this element of matter.
If r is the proper time, and x t ^(T) the space-time coordinates of this
material particle, we have
__ dx, __ pPUi _ / dx l
^l 7 ~> "j, ' 7
dr c c dr
We now choose the domain 1 in (97) as that part of this tube for which
r l < r < r 2 and consider an arbitrary variation of the time track satisfy-
ing the condition 8a\ - for r ~ r l and T -^ TO. Since
I
where dV is the rest volume of the particle, we have
r r t dx \
I /^j ^ uc 2 ) rfX I I A D dV ? -u rfFc 3 ) dr
J J \ < /T /
T> T.J
^ rf P f A t ^ </T -r, 3 ( /w f rf T . (101)
Tl TI
Here rfm = JLC rfF and rf^ - p dV represent the total rest mass and
total charge of the particle, respectively, quantities which are constant
along the tube.
From the definition of dr we get for the variation of the second integral
by a variation SO^T) of the kind considered
T 2 T 2
/ l /* 7 r T
S \ dr -= - ^ 8x L dr (102)
J <' 2 J dr
(see 91, p. 244).
Further, using
dr dr l ' dr dx k dr tix k A '
we get
5 f A ^ x ^ i f l^ A i ^ d> x i , A d8x t \ j
o \ A f ~ dr ~ ox,. - -4- A, dr
J l dr J \dx k k dr ^ l dr )
TI
f ldA kTT dA\ s , f l&A k tiA\ TJ
== I l -- U k -! o^.aT = (-7^ - ^IcAo^jaT
I \ /^'y* r/T / I \ f) i y f} f Y > i
j \ca^ ar / j \ox l o^kl
' JJ fir (]T (\(\ t \\
ik k i ' \ "'/
V, 59 ELECTRODYNAMICS IN VACUUM 159
From (98), (101), (102), and (103) we thus get for the variation of L in
this case
8L
r / J
= c J ^F lk U k -^
(104)
Thus the condition 8L = for any variation of the kind considered leads
to the equation i Tr
i.e. just the equations of motion (96) of charged incoherent matter.
60. The electromagnetic energy -momentum tensor
We shall now show that by virtue of Maxwell's equations the four-
force density (93) can be written in the form of the divergence of a
symmetrical tensor. Introducing (16) into the expression (93) we get
Further, we have
l8F
where we have made use of ( 1 3) and of the antisymmetry of F lk . Thus we
may write j i in the form ~
/. = -, ,,0.)
. ('06)
According to the rules of tensor calculus, S lA . is a tensor, it is symmetric,
S,* - -S*t, (107)
and it satisfies the identity
0. (108)
Using (11) in (106) we get by a simple calculation the following
expressions for the components of this tensor in terms of the electric and
magnetic field vectors: __ __/
where t lK = tf l tf (f +fl r ,#,-i(*+#)S w (HO)
is Maxwell's stress tensor; further,
160 ELECTRODYNAMICS IN VACUUM V, 60
where the vector S with components S L is Poynting's vector, i.e.
S = c(ExH). (112)
Finally, S 44 =-JF, (113)
where W = $(E*+H 2 ) (114)
is the electromagnetic field energy density.
The equation (105) with i = 4 may then be written
- *" x ij. , 1 dW
/4 = -(f.u) =a! --divS + s ,
(f.u)+divS + ^ = 0, (115)
ct
which expresses the energy conservation law if W and S are interpreted
as field energy density and current density, respectively. If we integrate
(115) over a finite volume V in space enclosed by a fixed surface or we get,
by means of Gauss's theorem,
-| J W dV - J S n da+ | (f . u) dV, (116)
V a V
where S n is the component of S along the outward normal of the surface
element da. The decrease in field energy inside V per unit time is thus
equal to the outward flux of field energy through the surface a plus the
total work done on the matter inside V by the electromagnetic forces.
For i = 1,2,3 (105) gives
f _ MI* 3 (S\ (
t'-wrwr { }
In the case of a static field the last term is zero and (1 17) is then exactly
Maxwell 's expression for the force density in a substance with = ^ =1.
The present deduction is due to Minkowski.f As pointed out by Abra-
ham, J the vector ^
g = | (118)
must be interpreted as electromagnetic momentum density if we want to
have conservation of momentum for a closed system. For if we integrate
(117) over the interior of a closed surface <r which contains the whole
system we get
f H. Minkowski, see ref., Chap. IV, p. 136.
t M. Abraham, Ann. d. Phys. 10, 105 (1903); Abraham-Becker, Theone der Elektn-
zitat, vol. II, 6th ed., Leipzig, 1933.
V, 60 ELECTRODYNAMICS IN VACUUM 161
since
/ p\ *
__i* dV by partial integrations can be transformed into an integral
J fa K
over the surface a where the field and therefore also t LK is zero. The left-
hand side of (119) represents the total force exerted on the matter and
is equal to the increase dG m /dt per unit time of the mechanical momen-
tum. Thus (119) may be written
In order to obtain a constant total momentum we have thus to assume an
electromagnetic momentum J g dv besides the mechanical momentum.
It is true that from this argument we can only conclude that the electro-
magnetic momentum density must be S/c 2 -f-(7, where C is a constant,
but since g must vanish simultaneously with the field the constant C
must be zero.
Now defining the velocity of propagation w of the electromagnetic
energy by the equation o
w= " w'
we can write for the electromagnetic momentum density
S 2 2
in analogy with (IV. 233), holding for the mechanical momentum
density. This analogy is close only in the case where the velocity w
defined by (120) satisfies the condition w < c. For a plane polarized
electromagnetic wave we have (E.H) = and E = //; thus we get
. IS| _ cEH _ _
i.e. the energy in such a wave travels with the velocity of light.
For the field of a charged particle in arbitrary motion we have, on
account of (61) and (62),
(E.H) = and E/H = <x^l,
cEH 2ca ^
i e w = __ -- <^ c.
^
61. The total energy -momentum tensor
If we use the expressions (105) for the electromagnetic four-force
density in the equations of motion (IV. 229), the laws of conservation
3595.80
162 ELECTRODYNAMICS IN VACUUM V, 61
of energy and momentum for the total system of matter and electro-
magnetic field take the form
| = 0, (122)
where T lk = O lk +K lk (123)
is the total energy-momentum tensor of the system. The components
of this tensor are obtained from the components of 6 lk and$,^ given in
51 and 60. We have
IK _ * /i 9j\
.^) ' (124)
where t iK is Maxwell's stress tensor.
where S = -^-T,. u 4 c(E X H)
,/(! u 2 /c~)
is the to\ 'il energy current density. Further,
(126)
is the total momentum density of the system.
Finally we have T 44 7/, (127)
where * =
is the total energy density.
The sum of the diagonal components of this tensor is, on account of
(108) and (IV. 198),
VI
GENERAL CLOSED SYSTEMS. MECHANICS OF
ELASTIC CONTINUA. FIELD THEORY
62. Definition of a closed system
IN Chapter V we have treated the case of incoherent charged matter
under the influence of electromagnetic forces. We saw that the four-
vector fi describing these forces could be written as the four-dimensional
divergence of a tensor which itself is a function of the field variables
describing the electromagnetic field. As the principle of relativity re-
quires that all signals are propagated with a velocity smaller than or
equal to c, it is impossible to maintain the Newtonian idea of forces
acting instantaneously over finite distances in space It seems necessary
to assume that all forces between material bodies are transmitted by
means of an intermediary field in a way similar to the case of electro-
magnetic forces. It is therefore generally assumed that all types of
forces can be described by a four-force density f t which is the divergence
of a certain tensor S lk depending on the field variables of the intermediary
fields by analogy with (V. 105). For the total system of matter and
fields we then get, in the same way as in 61, the Jaws of conservation
of energy and momentum in the form
'Ir^ ' (1)
where T lk is the total energy-momentum tensor of the closed system.
The physical meaning of the components T l4 and T 4l is the same as in
(V 125, 126, 127), i.e.
T* = -S (2)
4ti" * ^
where S is the energy current density,
T 14 = icg L , T 44 = -A, (3)
where g and h represent the total momentum density and energy density,
respectively. This should hold for any closed physical system, and hence
also for elastic bodies, if the elastic stresses and energies are included.
The equations (1) for i = 1, 2, 3 may now be written
which represents the law of conservation of momentum in differential
164 GENERAL CLOSED SYSTEMS VI, 62
form. T IK is called the stress tensor or the momentum current tensor
(cf 51, p. 137).
Similarly, the equation (1) with i ~ 4 represents the continuity equa-
tion for energy. ~ 7
divS + ^ = 0. (5)
ct
The total energy-momentum tensor of a closed system must be sym-
au - *i,. w
The spatial pait of this equation, i.e T LK -~ T KL , is essential for the
validity of the conservation law of angular momentum (see 51 and
f)3), and if this law is to hold in every system of inertia, the equation (6)
must also be valid for the space-time components, i e. we must also have
T A = T tl , (7)
or, on account of (2) and (3),
g - S/c* (8)
Defining the velocity of propagation u* of the energy by
u* -^ S/h (9)
as in the case of the electromagnetic energy (V 120), the equation (7) or
(8) can be written 7
which is formally analogous to the equation (IV. 233) for the mechanical
momentum density and thus shows that the energy density h corre-
sponds to a mass density 7/2
Tt should be remarked, however, that the velocity u* defined by (9) may
be larger than c and even if ?/* < c the transformation properties of the
velocity u* by Lorentz transformations will not in general be in accord-
ance with the transformation equations (l[. 55) lor the velocity of a
material particle. The energy density h may also be negative, thus
corresponding to a negative mass density /?/ 2 .
Introducing in every system of coordinates four quantities >V ? , where
we have, on account of (5), * ' (13)
c'.r,
in every system of coordinates The quantities >S t do not, ho\\ ever, trans-
VI, 62 MECHANICS OF ELASTIC CONTINUA 165
form like the components of a four- vector. Let us now assume that
h > and u* < c, i.e.
8,S t = S 2 -c% 2 <0. (14)
For every system of coordinates we may then define four quantities /7*,
analogous to the four-velocity of a particle, by
These quantities will transform like the components of a tune- like four-
vector when and only when the velocity u* transforms like a particle
velocity. We shall now find the condition to be imposed on the tensor
T lk for this to be the case. Since
77*
__
Now consider an infinitesimal Lorentz transformation
x( =~- x t + c lk x k , tk - - kl (17)
connecting the space-time coordinates of two systems of inertia S and S'.
From the transformation laws for a tensor \ve then get
thus, on account of (12),
' i I e ^' * + ,/**> A '
(17')
Expanding in terms of the infinitesimal quantities e tk we get, neglecting
terms of order higher than the first,
77*77 77*
, u i *ki u i
Thus, in order that U* shall transform like a four-vector the tensor T lk
must, for i ~ 1, 2, 3 and for all values of &, satisfy the condition
R lk ^T ik + T ^. J l = Q. (19)
166 GENERAL CLOSED SYSTEMS VI, 62
For i = 4 the equation (19) is identically satisfied. This condition is,
however, also sufficient, for, if (19) is satisfied in S, we have
I/?'-- U* + lK V* k , (20)
R lfi will then transform like a tensor by the transformation from 8 to 8'
and (19) will therefore also be valid in 8'. Since a finite Lorcntz trans-
formation may be composed of an infinite number of infinitesimal
Lorentz transformations, (19) represents the general condition which
T lk must satisfy in order that the velocity of the energy u* shall transform
like a particle velocity
In general, the energy -momentum tensor of a physical system will
not, of course, satisfy the condition (19) There are, however, a few
cases in which this condition must be fulfilled for physical reasons. In
the trivial case of a system consisting of incoherent matter without any
external forces, for instance, we have T lk = 9 1k ^U t U k , i e.
In this case the propagation velocity of the energy is, of course, identical
with the velocity oi the matter For an elastic body, however, the
condition (19) will in general not be fulfilled, as we shall see in 65.
In a later section ( 70) we shall meet another important case in which
the condition (19) must be satisfied lor physical reasons.
63. Four-momentum and angular momentum four-tensor for a
closed system
Putting (j, = l T, 4 T-. [g, l h\ (21)
ic \ c ]
the symmetry condition (7) may be written
17* -- !A (22)
c
and, on account <>t (13), we have also
(23)
t V '
It should be lemembcied that the quantities g t and 8 t are not four-
vectors.
Let us now assume that the system considered is finite so that all com-
ponents T lk are zero outside a certain region in physical space. If we
multiply (1) by (h\(Lr,djc 3 and integrate over the whole physical space
VI, 63 MECHANICS OF ELASTIC CONTINUA 167
for constant # 4 , the first three terms in (1), which are partial derivatives
of T IK with respect to the space coordinates X K , will give zero. Hence we
which shows that the four quantities
(24)
are constant in time G and // represent the total linear momentum and
the total energy of the system , respectively. Another consequence of
(1) is that the quantities G t transform like the components of a four-
vector. the four-momentum vector. This is seen in the following way.
Let a l be an arbitrary but constant four- vector. The four- vector
b k = a.T tk (25)
will then satisfy the equation
2- = - {26)
dx k
on account of (1).
If we multiply (26) by d^l - (Lr^djc^djc^djc^ and integrate over a
^
finite region D in (3 + l)-space, we get by means of the generalized Gauss
theorem (IV. 191)
0= f^rfS = [b k dV k , (27)
J ( * r A J
11
where Q is the three-dimensional boundary of the region H. Since we
have to deal with a finite system, the region in (3+ l)-space where T lk
is different from zero, i e. the time track of the system, will have the form
of a tube with a finite cross-section in the space-like directions
Consider two arbitrary coordinate systems 8 and $' in (3+l)-space
and two hyperplanes I2 t and 11, defined by the conditions
jc^ = constant
and 4 constant,
respectively, the values of the constants being arbitrary. For the region
X we now choose a domain bounded by the hyperplanes iij, } 2 and by a
cylindrical surface !3 3 enclosing the tube in which T lk ^ (Fig. 15)
The three-dimensional hypersurface Q, is thus composed of the parts
168 GENERAL CLOSED SYSTEMS VI, 63
I2 1? I2 2 , and (2 3 . The contribution to the integral J b k dV k from the cylinder
n
12 3 is zero, since T lk and b k are zero on Q 3 . Thus, we get
{ h dV k H f b k dV k =- (28)
i2i ila
The two integrals in (28) are invariant and may thus be calculated in any
system of coordinates We choose to calculate the first integral in S and
= const
- const
FIG 15.
the second in 8' If we suppose the events on D 2 to be later in time
than the events on 1^, the outward normal on I2j is pointing in the
direction of the negative time axis and, according to (IV. 191'), the
components of dV k are
dV, ~-- (0, 0, 0, 4-i dx* dx^dx^}
H \ 7. T, 7 \ 1 , J /
Thus we get
HI
where ,r 4 is kept constant in this integration
In the second integral we have similarly
hence
f r ' rt f r r r
h, dK. ^ b, dV i = i O 4 (.r 1? a*o, a
J A. K J h. /w J 4V 1' ->
Using (29) and (30) in (28) we get
6 4 dV =~ j b' 4 dV',
(30)
VI, 63 MECHANICS OF ELASTIC CONTINUA 169
or, by means of (25), (21), and (24),
a l G l = a\ G( = invariant. (31)
Since this equation must hold for an arbitrary constant vector a t it
follows that the connexion between G l and G\ is given by the transforma-
tion equation for a vector:
= ,*<?* (31')
For the validity of this proof of the vector character of G l it is essential
that T lk is everywhere regular. If T lk has a singularity along a certain
world line, we would have to exclude this line from the region S by a
certain surface 11 4 which then would give a contribution to the integral
fb k dV k m(21).
a
Since G l is a vector, G l G T is an invariant, and we may now define the
total proper mass J/ of the system by the equation
ff t O t =-Jfc 2 , (32)
by analogy with (IV 51), holding for a material particle.
By means of (1) and (6) we get further
A (XtTu-x^) - S f , ^-8^, - T kl -T lk - 0. (33)
dXj
Integrating this equation over the whole physical space we find by
arguments similar to those used before
which shows that the six quantities
M lk - J (x,g k -x k9l ) dV ~= -M k% (34)
are constant in time. This result depends essentially on the symmetry of
the energy-momentum tensor. By a method similar to that used in the
proof of the vector character of G l it now follows from (1) that the
quantities M lk transform like the components of an antisymmetrical
tensor: the angular momentum four-tensor with respect to the arbitrary
origin of the coordinate system.
The spatial part of this tensor M IK is, according to (IV. 98), dual to an
axial vector
(35)
which is equal to the constant total angular momentum vector of the
closed system.
170 GENERAL CLOSED SYSTEMS VI, 64
64. Centre of massf
We may assume that G t for any physical system is a time-like vector
so that M {} defined by (32) is a real quantity (cf 30) In this case it is
always possible to find a system of inertia $, the 'rest system', in which
the total linear momentum G -- 0, so that on account of (32) we have
for t he components of G L in 8 { \
G ( ! = (0,0,0,iJ/ c) (36)
Exactly as in the case of a material particle the velocity u of the rest
system ^S relative to *S is then
u -- c*G/H (37)
Tn Newtonian mechanics the centre of mass of a physical system with
the mass density ^ -- /u,(x,/) is defined as a point with the coordinate
vector , ~
X _l M (x,0 xrfr, (38)
\vheie M \ fidV is the total mass of the system In lelativistic
mechanics the mass density is connected with the energy density by the
equation (II) We ma\ then define the eentie of mass by the equations
(3<S) and (11) As we shall see in a moment, the point defined in this way
will, however, in geneial depend on the system of cooidmates which is
used m the evaluation of the integrals in (38), i e each system of inertia
A r has its own centre of mass (\tf) which depends on the system >V In /S 7
its coordinate vector X(G T (A>)) -- X(/S T ) is, according to (38), (11), and
(24), defined by
xrfl r (39)
From (23) we now get m every system of coordinates
' )(r/ ' tr * ) a 8 -a
> - ( Ji ik - :A^
( u ,
and by integiation ovei the whole 3-space
Since // is constant in time this equation for k -~ 1, 2, 3 shows that the
cent i e of mass defined by (39) is moving with the constant velocity
< 2 G/// re kit iv e to $, i e. with the same velocity u as the rest system S.
Thus all the different mass centres C(ti) aie at lest in the system $
f" \ 1) Kokkcr, Rcldtn itnt^thcoi u , (jionnigiMi, 1920, p 170, A Papapetrou, I'raktikci
\(fi>l <1 \thttnti, 14, ,~>4() (1{)3 ( )), (' Mollor, Cotnm Dublin lust for Advan< cd Studies,
Soi A, No .3 (1 ( )I9), () C <U* Heaunyaul, Kdatintt' tefit/etnte, rhap i\ , Pails, 1949,
M H L Ii>co, 7'/w Hoy Sot A, 195, 02 (1948)
VI, 64
MECHANICS OF ELASTIC CONTINUA
171
One of the centres of mass plays a distinguished role, viz. the point
C C(S Q ) which is the centre of mass in the rest system itself, it may
be called the proper centre of mass If X L (X, A" 4 ) are the space-time
coordinates of the proper centre of mass C (} in an arbitrary system of
coordinates, the X l X^T) will be linear functions of the proper time
r of this point. Further, if
U - dXl =
1 dr
denotes the four-velocity of (7? we have, on account of (37),
G,~M V t . (40)
The dependence of G l on the velocity of the proper centre of mass is thus
the same as for a material particle
Now denoting the relative angular momentum four-tensor m lk with
respect to the proper centre of mass C r by
'.-XJyL-^-X^] dV - M tk -(XJ! k -X k G,), (41)
we get, by differentiation with respect to the vanables .* 4 and r,
(42)
on account of (40). The two space vectors m and n defined by
m = (?/?,, Wai, W 10 ) \
-in -= ( f >,' u ! >C) )
are thus constants of the motion. We may therefore in (41) choose
x 4 X and, by means of (42) and (41), we get
((x-X)Xg)r/l'
J
ci-A'i
/ 7 y*
r J e a * ' ^ 4 Xt " J
(43)
Thus rn is the relative angular momentum vector with respect to the
proper centre of mass, the inner angular momentum, while n/c is the
moment of mass with respect to the same point Solving the second
equation (43) with respect to X we get
on
- l f
~>/J
hx dV
(44)
where X($) and X($) X are simultaneous coordinate vectors of the
centre of mass C(ft) in X and the proper centre of mass C($)
From (44) we see that the different centres will coincide only if n,
172 GENERAL CLOSED SYSTEMS VI, 64
and therefore m lk , is zero in every system of inertia, i e when the system
considered has no inner angular momentum If (44) is written in the
rest system xS we get
n -- or w? 4 = 0, (45)
since by definition X ----- X(tf). This condition is equivalent to the
covariant equation ,> n
{j '- u
as is seen from (36) when equation (46) is wiitten in the system iS' .
Thus this equation expresses in a co variant way that the proper centre of
mass is the centre of mass in its own rest system
When we choose the same orientation of the spatial axes in 8 as in tf
we get, by means of the transformation equations (IV. 81') for an anti-
symmetiical tensor, on account of (45),
where v u is the velocity of 8 relative to AS r and m is the inner
angular momentum vector in the rest system
The difference between simultaneous positions of the centre of mass
in 8 and the proper centre of mass is, according to (44) and (47), gi\en
by the time-independent space vector
a(,<7) - XOS')-X - -en/// .= (mxv)/3/ c 2 , (48)
where we have used the relations
following fiom (40).
Since the transformation from $ to Nis given by a Lorentz transforma-
tion without rotation, and since a is perpendicular to the relative velocity
v, the distance between the two centres, mentioned above, in the rest
system *S is also given by (48).
In the lest system $ all mass centres f(#) obtained by varying $ or v
in (48) foim a two-dimensional circular disk perpendicular to the
angular momentum vector m with centre at the proper centre of mass C
and with radius , ft!
" = v < 49 '
JU Q C
In the non-relativistic limit c -> oo the radius of the disk tends to zero and
we are left with one mass centre only, the Newtonian centre of gravity,
but in the relativistic case we have in general not one centre of mass but
the disk of mass centres mentioned above, the centre of which is the
VI, 64 MECHANICS OF ELASTIC CONTINUA 173
proper centre of mass. Only if the system has no inner angular momen-
tum is the radius (49) of the disk zero. It is true that the radius (49) for
all macroscopic systems is very small compared with the dimensions of
the systems For the earth, for instance, we have
/Varth - . 10 metrcs ' (50)
JM0C
For systems of atomic dimensions, however, the radius of the disk of
mass cent i es may be comparable \\ith the dimensions of the system.
Fiom the above considerations we can draw a certain conclusion
regarding the dimensions oi a system with given inner angular momen-
tum m and proper mass M {} Consider an arbitrary physical system
which in the rest system 8 lies entirely inside a sphere with centre at
the proper centre of mass C and radius r, i.e. a system for which aU
components of the energy -momentum tensor are zero outside this
sphere. If we further assume that the energy density h is positive every-
where in all systems of inertia, it is clear that the whole of the disk of mass
centres must lie inside the sphere; for if we consider an arbitrary point,
say C(R), on the disk, this point will in the system of coordinates H be a
centre of mass, and since h is positive it must then lie inside the physical
system We thus get , ,
r >> 'J5 ' (51)
3/ c
Thus, a system with positive energy density and u ith a given inner angular
momentum m () and a given rest mass M (} must always have a Jinite extension
in accordance with (5 L) It the system is smaller, h cannot be everywhere
positive in all systems of inertia
65. The fundamental equations of mechanics in elastic continua
Jn ( Chapter IV, 50 51 , we have treated the mechanics of incoherent
matter under the influence of given external forces. We shall now con-
sider the case of an clastic body with no external foices The sole forces
acting in the body urc then the clastic torces between neighbouring
parts of the matter due to the deformation of the matter. We have thus
to deal with a closed system \vhieh is a special case of the general
systems considered in 62, and the equations (l)-(l 1 ) must be valid for
the total energy-momentum tensor T lk of this mechanical system. The
mechanical energy-momentum tensor has, however, especially simple
properties which we shall now establish
(Consider an infinitesimal lace element da with a directed normal
defined by a unit vector n, at a definite point p in space The matter on
m
GENERAL CLOSED SYSTEMS
VI, 65
either side of this face element experiences a force which is proportional
to da. The force acting on the side to which the normal points will be
called t(n) da, and, since action and reaction are equal, the force on the
other side t( n) da must then be t(n) da If n (1) , n (2) , n (3) are unit
vectors in the directions of the Cartesian axes, we have
t(n) =
(52)
where w 1? ?? 2 , n% are the components of the unit vector n The equation
(52) is obtained by a consideration of the infinitesimal piece of matter
Fio 16.
contained in the pyramid abcp of Fig 1 6 If da is the area of the triangle
abc, the area of the triangles pbc, yea, and pab are n^da, n^da, n^da,
respectively, and the total elastic force on this piece of matter will be
-t(n)
rf<j-ht(n< 3 >)w 3 da
This force must be equal to the change of momentum of the matter per
unit time, i.e. d(&oV)/dt, where g is the momentum density and oV
the volume of the pyramid In the limit of an infinitely small volume,
oV tends to zero faster than da, therefore
VI, 65 MECHANICS OF ELASTIC CONTINUA 175
which leads immediately to the equation (52) If the components of the
vectors t(n (K) ) are denoted by t iK , (52) may be written
t,(n) - t lK n K (53)
Since t L (n) an( i n K are the components of space vectors, the quantities
t lK must transform like the components of a space tensor by rotations of
the Cartesian axes. t LK is the elastic stress tensor, sometimes called the
relative stress tensor, in contrast to the space part T LK of the total energy -
momentum tensor T lk which is called the absolute stress tensor |
The total elastic force F acting on the matter inside a closed surface
a is now equal to
- Jt(n)rf<r,
a
where n is the outward normal to the surface element r/cr.
The components F L of this force may, by means of (53) and Gauss's
theorem (IV. 192), be written
l(t k rf ff = - j^'dV, (54)
a Si
where the integration on the right-hand side is extended over the interior
of the closed surface a Thus \ve can define an elastic force density f
such that
*',- //,</!', (55)
1*2
and a comparison of ( r >4) and (55) shows that the clastic force density
and the relative stress tensor are connected by the equation
/.= -- *)
The motion of an infinitesimal piece of matter with the volume 8V
is now determined by the equations of motion
^Sn^Sr.---^!', (57)
where g is the momentum density and d/dl denotes the substantial time
derivative By means of (IV 201) and (IV. 204) we get
- 68 >
f M \on Lane, Die Relativitatsthcone (3rd ed , Braunschweig 1919), vol i, 29;
Ann d Fht/a 35, 524 (1911).
176 GENERAL CLOSED SYSTEMS VI, 65
where the U K are the components of the velocity u of the matter at the
place considered. From (57) and (58) we then get
JLi-j (g^tc^uc) 0. (59)
On the other hand, the law of conservation of momentum is also
expressed by (4); thus we obtain the following connexion between the
absolute and the relative stress tensors
T LK = t lK +g L u K . (60)
In order to find an explicit expression for the momentum density we
shall use the connexion (8) between g and the energy flux S:
g - S/c 2 . (61)
The total work done by elastic forces on the matter inside a closed
surface a per unit time is
A --. f (t(n).u)dcj = -
a a 11
where the integration in the last integral is extended over the interior 12
of the surface a. The work done on an infinitesimal piece of matter of
volume W is thus f]( . ^
8 A = _ iT-'^sr. (62)
This must bo equal to the increase per unit time of the energy inside SF
which is
- K -~ K , (63)
rfr *' v '
/ being the total energy density including the elastic energy. Thus we
get from (63) and (02)
? h i + ,?-(l + vJJ = 0. (64)
01 (.1 K
A comparison of (5) and (64) shows that the total energy flux is given by
S-Auf(u.t), (<> r >)
where (u t) is a space vector with components (u.t)^ --- u L t lK Thus,
besides the convection current hu there is an extra transport of energy
due to the work done by the elastic forces From (61) and (11) we then
get for the total momentum density
g_ _. + ("'. ,66,
VI, 65 MECHANICS OF ELASTIC CONTINUA 177
where /x = h/c 2 is the total mass density including the mass of the elastic
energy On account of the last term in (66) the momentum density
vector has not in general the same direction as the direction of motion
of the matter . hence ,
9i U K-9* u i
Since the law of conservation of angular momentum requires T IK ----- 7^ t
(see 63), we get from (60)
IIK-IKI ^ ~</c U K \^K U c = (-(U.t) L U K +(U.t) K U L )/C* ^ 0,
i e the relative stress tensor is not symmetrical
Only in the momentary rest system A^ of the matter at the point con-
sidered, we have u = and thus, on account of (60), (65), and (66),
/O _ /po ._ 7H) __ /O tfO __ dO rfiQ _ M 710 . _7,0 (l\>7\
l iK -- -*- IK HI 1 KI> *\ fel - - L 14 - U > L 44 ~ II 9 V' I
where h is the rest energy density.
The mechanical energy-momentum tensor satisfies the equation
T tk T? k = -A U t , (68)
where I \ is the four-velocity of the matter The validity of the co variant
equation (68) follows at once from (67) if it is written down in the rest
system, where U** = (0, 0, 0,tr). The relation (68) is characteristic of
a pure mechanical energy-momentum tensor and contains equation
(IV. 246) as a special case. If we multiply (68) by U t we get the follow-
ing expression for the invariant rest energy density
A ^ l^T lk U k jc*. (69)
For & = e ~ 1, 2, 3, (68) gives
or, by means of (60), (3), and (TV 39),
(^+^vK-c 2 t -- -AX.
Solving with respect to (J L we get a new expression for the momentum
^u + (l/e*)(t.)
& 1 9 r~ ( T - " i \ ' u /
where /x = h Q /c 2 is the rest energy density and (t. u) is a space vector
with components t llc u K . By means of (IV. 39), (70) may also be written
-- 4 . (71)
The equation (68) for i 4 gives similarly, on account of (70),
/z, -j-(u.t. u)/c 2
with (u.t.u) = u L t iK u K .
3595.60 XT
178 GENERAL CLOSED SYSTEMS VI, 65
(72) may also be written
T u ---- -h - + M C/ 4 U t - l fS V^ U K . (73)
The equation (69) gives similarly
.u), (74)
r-
which is identical with (72) The right-hand side of (74) is thus an
invariant scalar
If we divide (74) by c 2 we get
which is a generalization of the equation (IV 199) valid for incoherent
matter
Comparing the two expressions (66) and (70) for g we get, by means
of (75), the following identity for the relative stress tensor t lK
(l-w 2 /<' 2 )(ii.t) . (t.u)-u(u t.u)/r 2 (76)
A closed system may always be divided into non-closed sub-systems in
an infinite number of ways corresponding to a division of the energy-
momentum tensor into separate parts. For instance, we may write
where B, k ^U t V k (7S)
is the kinetic energy-momentum tensor By means of (60), (71), and
(IV. 39) we get
Thus we get for the components of the tensor K,^ from (70), (71),
and (73)
' " ,, a
(^ o __ rjJ T T TT t.A A 4
(SO)
On account of (68) the tensor ^ A satisfies the condition
a relation which also follows directly from the expression (HO).
In the rest system we get from (80)
-SSc - C, '^4 - ^, -= (82)
VI, 63 MECHANICS OF ELASTIC CONTINUA 179
Putting /, olrtst = aSffA , (83)
dx k
the equations (1) may be written in the form
* - /, elast
analogous to (IV 229) Tt should be noted that the force density
f' last , defined by (S3), is not in general identical with the elastic force f
defined by (56) A simple calculation shows that (// hlst f / ) - which, as
shown in 50, means that the proper mass of the system is not conserved
This is also natural, since /x, includes the mass coi responding to the elastic
energy which changes under the influence of the elastic forces While
the tensor O lk satisfies the condition (19), the total mechanical energ\ -
momentum tensor T tk will in general not satisfy this condition
The relative stress tensor t LK is connected with the internal deformation
of the matter In the rest system /S T this connexion is given by the
equations of the usual theory of elasticity, thus for small deformations
it is given by Hooke's law By means of the transformation properties
of the stress tensor this connexion can be established in any system of
ineitia |
If the elastic body is subjected also to external non-mechanical foices
described by a four-force density f[ xt w r e have instead of (1)
(S5)
or by means of (77) and (83)
66. Transformation of elastic stress, momentum density, and
energy density
Let us assume the spatial axes in the coordinate systems N and iS to
-have the same orientation. Since the velocity of >S relative to *S is the
same as the velocity u of the matter at the point considered, the trans-
formation coefficients a lk are given by (IV 125), i e
/i/ rti i
(87)
t G Heiglot/., Ann d I'lnjs. 36, 493 (1911).
180 GENERAL CLOSED SYSTEMS VI, 66
From the transformation equations of a tensor in connexion with (67) we
gCt T tk -, T*^*^ -~ f^w^-h'w*. (88)
For i k 4 this gives
(89)
A comparison of (89) and (72) shows that we must have
(u.t.u) - (u.t.u). (90)
For i --= i -- 1, 2, 3, k ^ 4, we get from (88), by means of (87),
or, in \ector and tensor notation,
In the same way we get from (88) with i c, fc -- K-, and from
(<>0) and (1)1)
(93)
Intiodueing the notation a O b for the direct product of the space
vectors a and b, \vhich is a tensor with components a L b K1 the formula
(93) may also be written in thiee-dimensional tensor notation
U- yu U y
(94)
By means oft/his tiansformation equation it is easily verified that the
T elation (90) is valid, further remembering that t (} tK is symmetrical, i e.
(t . u) --- (u . t), a simple calculation shows that the expression (94) ior t
is 111 accordance A\ith the equation (76).
In the special case where u (u,0, 0), i.e. where the motion of the
matter at the point consideied is parallel to the #-axis, the transformation
VI, 66 MECHANICS OF ELASTIC CONT1NUA
equations (89), (92), and (93) reduce to
181
1
i -* i p i Q r _ iQ
yjc v ** ' i/// 'UU^ U** ' UZ
7
y ' ~" ""'
67. Perfect fluids
In a perfect fluid the force t(n) on a face element \\ith normal n is
parallel to n, i.e.
t(n) _
where p is the normal pressure Thus we get from (53)
where p is the normal pressure.
In the rest system we have in particular
The transformation equations (93) in this case i educe to
P ^LK ~~ P^ ^IK>
or P ~~ P>
i e the normal pressure is an invariant scalar f
From (SO) and (97) we now get for a perfect iluid
(100)
i e
(101)
(102)
where ^> - p Q is the invariant pressure. This expression for 8 lk also
follows from the fact that (102) is identical with (82) in the rest
system.
t M. Planck, Berl. Ber , p. 542 (1907), Ann. d. Fhys 76, 1 (1908).
jH2 (;KNKK\L CLOSKD SYSTKMS vi, ^GT
From (102) it follows that the piessure p is one-third of the invariant
diagonal sum of the tensor >V, A , i e
v - J'S, ( lo:J )
The total eneigy-momentum tensor toi a perfect fluid is then, by (77)
and (78),
The pressure /> />" is a Junction of the rest density jj? and the temper a-
tuie which j,s given by the equation of state of the fluid.
It the fluid is subjected to external forces with the foiu -force density
/ t rxt , the ecjuations of motion ot the fluid may be written in the form (86)
OJlk C CXfc C * k * l -. (105)
r 2 ( ] x k r 2 dr r 2 dr Pjt\
Let us assume i // >xt to be of the type (IV. 214) which satisfies the
identity (IV. 216), P,/;' xt The action of these forces will then not
give rise to an\ creation of pioper mass It is different, however, \Mth
the forces (105) ior which
<)jc k
= V *- -- ^dtv u (106)
div u is the \ olurne dilatation in the rest system, j)div u repre-
sents therefore the inciease in elastic potential energy density per unit
time in the rest system. JSince ^ includes the mass corresponding to
elastic potential energy, ' A thus represents the rate of creation of
c- 'ojc k
proper mass density in the icst system Multiplying (8(5) by U t and sum-
ming we now get the equation analogous to (IV 226)
\ o : o 5 V/
dx k c 2 c 2
f~. rj
which just expresses that
c 2 f dx k
represents the source density of proper mass. Only if the fluid may be
considered as incompressible do we have dU k /dx k div u 0, and the
proper mass will be conserved.
VI, 67 MECHANICS OF ELASTIC CONTINUA
By means of (107) we now get
/Ti ('Ei (*>T <
183
(109)
Thus, using (109) and (105) in (86), we get the following equations of
motion for a perfect fluid.
.
C 2 dr
(no)
^ }
For the energy density and momentum density we get from (66), (70),
(89), and (97), or directly from (104),
(in)
If /x, ^, and u are constant throughout the elastic body, we get by
integration of (111) over the whole volume V = F\/(l ^ 2 /c 2 ) of the body
u
G - gF =
1
(112)
From these equations we see that the total momentum and energy,
i e. the quantities (G, (i/c)//), do not form a four-vector in this case.
This is not in contradiction with our general result in 63, because the
system is not a closed system. In order that the quantities ^t , />, u can be
constant throughout the body, the fluid must be contained in a vessel,
the walls of which will act on the system with forces which are not
included in the energy-momentum tensor (104). (See Chapter VII.)
From (112) we find, however,
G -**?.,.
C*
II+pV =
(113)
which shows that the system has the same momentum as a particle with
energy E H+pV and rest energy E = H+pV. The quantities
(114)
thus transform like the components of a four-vector.
184 GENERAL CLOSED SYSTEMS VI, 68
68. Scalar meson fields. General field theory
While the iorces between the atomic nuclei and the outer electrons
are properly described by electromagnetic fields, the characteristic
short-range property of the forces between the constituent particles of
the nuclei indicates that the nuclear iorces are of an essentially non-
electromagnetic nature. In order to account for the nuclear forces
Yukawaf introduced the so-called meson fields. The simplest type of
meson field is the scalar field described by an invariant scalar field
function x F(.r t ) satisfying the field equation
d 2l F
-K 2 T--0 or
W*
Here K is a constant connected with the range ol the nuclear forces and
in the case of a 'neutral' meson field X F is a real function of the space-
time coordinates (jr t ).
<F
Introducing the notation X F, ^ - ,
ar t
the field equations can also be written
^YL- K V\' (116)
These equations may be derived fiorn a vanational pimciple
8 J i(T, TJ dZ == 0, (117)
where il ^ -i^F^-f^Y 2 ). (118)
In fact, if the variation S X F -= ST^rJ of X F is assumed to vanish at the
boundary of the arbitrary four-dimensional region of integration we
have, since S X F,
(119)
Now, as this e\})iession is to be zero for any vaiiation of X F of the type
considered, we get
. f S r/X - r f SiWi; = f (^ 8 X F+ ^ 3 X I^ dX
J I J \0T f^ /
If
which are the Euler e(]uations corresponding to the vanational principle
(117). With the expression (118) for AJ (1^0) is identical with the field
equations (116).
t H Yukawa, P?oc Math.-P/ty*. Soc Japan, 17,48(1935)
VI, 68 MECHANICS OF ELASTIC CONTINUA 185
The energy-momentum tensor of the scalar meson field is given by
T tk = T/F.+fi 8,, = f,H;-!OT+KF') 8,,. (121)
On account of the field equation (116) this tensor is easily seen to satisfy
the equation ~m
~ 7 -* = (122)
dx k
holding for a closed system.
The scalar field is a special case of a general field described by a
number of field vanables
Ql^Qlfa) - (#Vi), Q*(* t ), -). (123)
Suppose that the field equations aie derivable Ironi a variational
principle
SJfirfS-O, (124)
where U --- A!(<^, Q\) (125)
is a certain algebraic im ariant function of the field variables and their
first derivatives ~ n r
(^ - d( . (126)
1 d*i
This means that the field equations are of the form of the Euler equations
o n-> 7 )
1 - 7
following fiom the vaiiational principle (124). Since is su])posed to
be an invariant, the equations (127) will have the same form in every
system of inertia
In virtue of the field equations (127) the quantity
+8 '* (128)
is now easily seen to satisfy the divergence relation
gi-* = (129)
In fact, we have
8x k ' bQl ^ ar, dQ l ftr
QQ^ fjQ^,
on account of the relation - - = - ~ following from (126).
b
186 CiKNKKAL CLOSK1) SYSTKMS VI, 68
The quantity (128) may therefore be taken as the energy-momentum
tensor of the field.
When iJ is an invariant, 6 lk is easily seen to be a tensor. It is called the
canonical energy-momentum tensor, its time component T 44 being equal
to , where
is the Hamiltonian density
In the case of the scalar field, the expression (128) for 6 lk reduces to
the tensor T lk given by (121). In general, however, 0, A will not be sym-
metrical and O lk will differ horn the real energy-momentum tensor by a
divergence-free tensor / iA , so that
where t,,-t u -, -(0,,-0J, '/ =,- (132)
( *k
Belmfantej and RosenfeldJ have given a general formula for the
calculation of /, A .
As another example we consider again the case of an electromagnetic
field in vacuum treated in Chapter V In this case the field variables
Qt are the components of the four-potential A k , and the function ii is
fi =-= \F lm b] m -
V A l
where A lm ;-- / .
"''-tn
The Kuler ecpiations (127) then take the form of the Maxwell equations
(V 10) in vacuum
-(.4,-^ --^-0 (134)
or r
Together with the Lorentz condition (V. 22) which must be regarded as
an accessory condition, this gives the wave equation
t F ,T. Bobnfanto, Fhi^ica, 6, SH7 (1U39), ibid 7, 30,"> (1940).
J L KuM'nfold, Mcinvirctt dc rAiail. Roy Bclyique, 6, 30 (1940)
VI, 68 MECHANICS OF ELASTIC CONTINUA 187
The canonical energy-momentum tensor (128) is now
0,* - I F u ^-mM 8,1 (135)
This tensor is not symmetrical and it deviates from the symmetrical
electromagnetic energy-momentum tensor (V 106) by the term
',. - -A A (136)
which satisfies the equation
a/ = o
ftr A .
on account of (134) and of the antisymmetry of the tensor F kl .
VII
NON-CLOSED SYSTEMS. ELECTRODYNAMICS IN
DIELECTRIC AND PARAMAGNETIC SUB-
STANCES. THERMODYNAMICS
69. General properties of non-closed systems
A CLOSED system H may be divided in many ways into two non-closed
systems X (l) and S (2) corresponding to a, decomposition of the total energy-
momentum tensor T lk into two parts
In the case of electrically charged matter, T$ may, for instance, be the
mechanical energy-momentum tensor and T$ the electiomagnetic
tensor. Defining a four-vector f l by
we get fioni (1) and from (VI. 1),
wj?
BV, Jl ^
f t is the four-force density produced by the system H (2) and acting on the
system (1) . Thus the fundamental equations of a non-closed system are
of the form ~ m ^ ct
-'--!?
where f t is the force acting on the system with the tensor T lk . The force
acting on the system with the tensor N |A is then f t .
The physical meaning of the space-time components T^ and T it of
the energy-momentum tensor of the non-closed system is as in (VI. 2, 3),
i.e. T ti =-. -*,<?,, T lt - icg., T M - -h\
(3)
wheie />, S, and g now denote the energy density, energy flux, and
momentum density, respectively, of the non-closed system.
Instead of (VI. 4) and (VI 5) we now have
S = - C / 4> (4)
4 v '
VII, 69 ELECTRODYNAMICS, THERMODYNAMICS 189
which represent the momentum and energy theorems for a non-closed
system, by analogy with (IV 238, 236)
In \iew of the arbitrariness in the decomposition (1), the energy-
momentum tensor of a non-closed system need, however, not necessarily
be symmetric, but in any case we must have
T tle -T lt --= -OS'^-A,). (5)
The total linear momentum and energy
of a non-closed finite system is, of course, in general not constant in time.
Integrating (2') over the whole physical space in an arbitrary system of
inertia /S, we get
If G L (t) and G' L (t f ) represent the momentum and energy of a non-closed
system in two different systems of inertia, the connexion between Q l
and G[ will not be given by (VI. 31') This is already obvious from the
fact that there is no unique connexion between the variables t and t*
occurring as arguments in G l and G(. But, even for a stationary system
where G t and G[ are time-independent, the quantities G l will not trans-
form like the components of a four-vector (see 70). This follows at once
from the proof given in 63 for the vector character of G l in the case
of a closed system For a non-closed system we would get, instead of
(VI. 28),
n,
and the right-hand side of (7) will be zero only for very special non-
closed systems.
Also the angular momentum defined by
W (8)
will now be time -dependent. From (2') we get, instead of (VI. 33),
3( x i T kl~ x k T il) __ f f im rp
~~~ - p~~ -- ~~ * Jk~~~ x k Ji i J-ki *ik'
OX^
Hence, by integration over the whole physical space,
| J/ lfc = J (x t f k -x k /.+T ta -5T lfc ) dV. (9)
190 NON-CLOSED SYSTEMS VII, 69
Thus, in this case, the density of the moment of the forces has to be
defined by
d lk - *, A-ai/.-f T kl ~-T lk _=. *, A-J-, A+^-SV (10)
For a non-closed system the centre of mass loses its physical impor-
tance Defining the coordinate vector of the centre of mass in the system
of inertia ti by the equation (VI 39)
X(A')- ~J/>(x,o xrfr- 1 f x</ 4 rfr, (ii)
we get, by means of (9) with i -- = i, I- 4 and of ((>) with i 4, after
a simple calculation,
(12)
The velocity of the centre of mass is thus not equal to c 2 G/H as for a
closed system even if the energy-momentum tensor is s\ mmetncal This
severely limits the value of the centre of mass as a representative 4 point
of the physical system.
In a closed system the pjo/tet centic of HI aw \\ us the centre of mass in its
own rest system We may now also for a non -closed system try to define
a representative point inside the system which at any time is the centre of
mass in its momentary rest system, the rest systems being, of course,
different at different times A closer investigation shows, however,"]*
that the representative point is not uniquely defined by this condition.
In fact, even in a closed system there is an infinite number of points
which at any time are centres of mass in their momentary rest systems.
For if we imagine the disk defined in 64 to rotate with constant angular
velocity . , 2
in the rest system S of the proper centre of mass, any point on the
rotating disk will be the centre of mass in its momentary rest system ( Con-
sider, for instance, a point p which at the time considered has the radius
vector a reckoned from the centre of the disk Its \ elocity is then
yir r 2
v - (o> x a) =--- - - (m X a), (14)
m^ v v 1
hence -ff- ^ _ (mx(mxa)) = a. (15)
iw e- |m |
t C. Mollei, Ann Inst Henri I'oiniate, 11, fast- v, 251 (1950).
VII, 69 ELECTRODYNAMICS, THERMODYNAMICS 191
A comparison of (15) and (VI. 48) shows that the point p is centre of
mass in a system j$ moving with the same velocity v relative to 8 as
the point itself, i e. any point on the rotating disk is centre of mass in its
own rest system.
In the case of a closed system it was possible, however, to single out
one point, the proper centre of mass, by the condition (VI. 40), which
means that the total linear momentum of the physical system is zero in
the lest system of the point This is not possible in the case of a non-
closed system For if we apply the equation (12) in the momentary
rest system of one of the representative points defined above, the left-
hand side is zero and the equation ( 1 2) then shows that the linear momen-
tum will, in general, not be zero in this system of inertia, and even if
this should be the case at the moment considered, it w r ill not be so at a
later time. Thus a unique generalization of the Newtonian centre of
gravity for non-closed relativistic systems is possible only for very special
external forces (see 70) There is one important exception, however,
as we shall see in Chapter X, 114 Tf the ext ernal forces are gravitational
forces, and if the system is sufficiently small, it is always possible to define
uniquely a proper centre of mass with all the properties of tho Newtonian
centre of gravity
70. Static non -closed systems
Let again T lk be the energy-momentum tensor of the system con-
sidered but let/, now be the four-force density produced by the system,
then, according to (2'), we have in every system of inertia
The system is called static if a system of coordinates ti exists in which all
physical variables are time-independent and if, further,
G = J g r/r = [ S dV - (17)
The system as a whole is therefore at rest in the system $, and since all
physical variables are time-independent in $, it is clear that also the
centre of mass in 8, as defined by (11 ), is at rest in 8 The system con-
sidered thus represents a case in which an unambiguous generalization
of the Newtonian centre of gravity is possible for a non-closed system.
As an example we may think of the electromagnetic field of charged
matter at rest in a definite coordinate system $. The tensor T lk is then
the electromagnetic energy-momentum tensor 8 lk which for a substance
192 NON-CLOSED SYSTEMS VII, 70
with e = /LI =^ 1 is given by (V. 106), f L being then the electromagnetic
four-force density acting on the charged matter.
Another simple example of a static non-closed system is a fluid con-
tained in a vessel under the influence of the external pressure from the
walls of the vessel.
To find the total energy and momentum in a system of inertia 8 with
respect to which S Q is moving with the constant velocity u we may use
the transformation properties of a tensor and the expressions (VT. 87)
for the transformation coefficients a lk .
Integrating the equation
^---Tl iah a mk (18)
over the whole space, we then get, using the Lorentz formula
dV --
and (17),
G -
- f gr/r ^ U
r<w.u
(19)
where T is the spatial tensor with the components T K Although G
and H are constant in time they do not transform like the components of
a four- vector In general, this may be taken as a proof that the system
considered is non-closed. For an elastic body, the equations (19) are
obtained from (VI. 92, 89) by integration, if we assume that the velocity
u is constant throughout the body. Such a system cannot therefore be
closed unless the stress tensor t -- T everywhere in the body.
If T is of the form ^ __ <> , m
* IK P LK> \^W
(19) reduces to the same equations as for a perfect fluid, i e. (VI. 112).
71. Electrostatic systems. Classical models of the electron
Let us now consider in a little more detail the case of charged matter
at rest in a system of coordinates S. If - ^ 1, the tensor S lk is
given by (V. 106 114). Since the field is electrostatic in 8 we have
H = 0, and E is constant in time, i e.
- 0, S K - -EE + l\E\* S iK . (21)
VII, 71 ELECTRODYNAMICS, THERMODYNAMICS 193
Let us in particular consider the case of a spherically symmetrical distri-
bution of the electric charges. In this case the field will also be spherically
symmetrical, E being directed along the radius vector connecting the
centre of the charge distribution with the point considered. Hence
and J $' dV ^- 1 6 j |ET dV 8 llt = i J A dV S, K = Jffo 8 t(t (22)
Using (22) m (ID) we get for the total electromagnetic momentum
and energy of a spherically symmetric charge distribution
r el
G( " = 3 7* V(i~*/<- 2 l' Ol " vl
Such a system represents a classical model of the electron, the funda-
mental equations of Lorentz/s electron theory being identical with
Maxwell's equations for substances with e = /t 1. Lorentz put for-
waid the idea that the mass, energy, and momentum of the electron could
be of purely electromagnetic origin, but from (23) we see that this is
impossible, f since the dependence of the electromagnetic energy on the
velocity differs from the relativistic formula (III 31) for the energy of a
particle. Since the quantities (G el , (i/c)H^) do not transform like the
components of a four-vector, we have to deal with a typically non-closed
system, and in order to get a consistent classical picture of the electron
we must assume the existence of non-electromagnetic energies and
momenta inside the electron at least as long as Maxwell's equations are
supposed to hold throughout the whole space.
Let us now assume that the charge e is uniformly distributed over the
surface of an elastic sphere of radius a in the rest system. If n is a unit
vector in the direction of the radius vector, the solution of Maxwell's
equations is
E = - n for r > a, E = for r < a, H =- 0, (24)
J
where r is the distance from the centre of the sphere. Thus we get
from (21)
l n L n K J ri . 2 -4 S t *< for r >
\E\ 2 dVO^
/ n * n i n * \ /
r O 1 j
el~ij
a
t M Abraham, Phys. ZS 5, 576 (1904)
3695.80
194 NON-CLOSED SYSTEMS VII, 71
Here the charge e, is measured in Heaviside units and m^ is the electro-
magnetic contribution to the rest mass of the particle.
According to (V 1 09) the electric force per unit surface on the sphere is
which must be in equilibrium with the elastic force Thus the elastic
stress tensor inside the sphere must be of the form
where p Q = --- , , = ~--^\ (28)
1 24 3
2(47r) 2 a 4
The total mechanical energy and momentum can now be obtained from
(VI 112)
u / _ 1 7/SA u
(29)
ff _ mo -
'"" " (T-^Va) " """"(l-^c 2 )" ""
Adding the expressions (23) and (29) we get for the total energy and
momentum
/"^ /^ I (^
- im--t- '!
i/ F/ I //
''
as we should have for a closed system. A system of that kind was used
for the first time by Pomcare as a model of the electron | Poincaro did
not specify the nature of the forces which in his model counterbalance
the electric forces in the electron ; he simply assumed the existence
of such forces of non-electromagnetic nature and a corresponding
energy-momentum tensor which together with the electiomagnetic
tensor defines a total energy-momentum tensor T lk satisfying the con-
dition dT lk /d.r k 0, characteristic of a closed system
In contradistinction to this duahstic point of view, which requires the
introduction of field quantities of a non-electromagnetic nature, MieJ
and Born advocated a unitary point of view in which only electro-
magnetic field variables are introduced These field variables must then
satisfy equations which deviate from the Maxwell equations inside the
t H. Pomcare, Rend Pal. 21, 129 (1906)
t G Mie, Ann d. Phijs 37, Ml (1912), 39, 1 (1912), 40, 1 (1913)
fc M liom, /Voc Roy Hot A, 143, 440 (1934)
VII, 71 KLKCTRODYNAM1CS, THERMODYNAMICS 195
electron where the field is strong These field equations are non-linear
and the corresponding energy-momentum tensor S lk satisfies the neces-
sary condition dti lk /dx k =- 0, i e. the self-force / t = dS lk /dx k is zero.
The final solution of the problem of the electron and of the other
elementary particles can probably not be found on a classical basis.
Besides the introduction of Planck's quantum of action it may even be,
necessary to introduce a new fundamental constant of the dimension of a
length f But the above considerations show that, as long as one assumes
the existence of an energy -momentum tensor of the system, the theory of
relativity requires the vanishing of the self-force, i e of the four-dimen-
sional divergence of this tensor
72. The fundamental equations of electrodynamics in stationary
matter
As shown by Loreiitz,f Maxwell's phenomenological equations of
electrodynamics for stationary matter may be derived from the funda-
mental equations of the electron theory by avei aging over regions in
space which are small from the macroscopic point of view, but still so
large that they contain a large number of electrons. Since the equations
(V 1 3, 1 0) of the election theory are covanant in for m, it must be possible
also to find the 'macroscopic 1 equations of electrodynamics in moving
bodies by averaging oxer appropriate small space-time regions This
was actually done b\ Born and Dallenbach |j
However, if we assume the validity of Maxwell's phenomenological
equations ior a body at rest, it is possible to find the corresponding equa-
tions in moving bodies simply by performing a Lorentz transformation
This method was used for the first time by Minkowski. j| The principle
of relativity requires ]bhat Maxwell's equations for stationary matter
must hold in that system of coordinates S Q in which the matter is at
rest, irrespective of the velocity of this system with respect to the fixed
stars. Thus we have in 8
curl E + ~ - = 0, div B -
: (3D
curl H - - D = J/r, div D
c
t W Heisenbeig, Ann d Phys 32, 20 (1938)
| Soe rof , Chap I, p 21.
& Almkow ski Horn, Math Ann 68, f>2G (1910), soo also A D Fokkor, Phil. Mag.
39', 404 (1920)
H W Dallonbach, DJSS , Zuiich, 1918, Ann d Phij? 58,523(1919)
ft Minkowski, H , Gott Nachr , p 53 (1908), Math Ann 68, 472 (1910)
HHj NON-TLOSKI) SYSTEMS VII, 72
where E, D, H, B denote the electric field strength, electric displace-
ment, magnetic field strength, and magnetic induction, respectively.
p and J are the macroscopic charge and current densities All these
quantities may in principle be determined by means of macroscopic
experiments in *S E and D, for instance, are defined as the forces on
a small test body of unit charge inserted at the point considered into
small crevasses cut in the matter parallel or perpendicular to the field,
respectively Similarly, H and B are the corresponding forces on a
magnetic test body of unit magnetic pole strength.
Besides the field equations (31) we have in isotropic dielectric and
paramagnetic substances the following constitutive equations connecting
the field variables with the constitution of the matter
D - eE, B /xH , J - aE, (32)
wheie is the dielectric constant, ^ the magnetic permeability, and a the
electrical conductivity The last equation is the mathematical expres-
sion of Ohm's law
73. Minkowski's field equations in uniformly moving bodies
Consider two antisvmmetnral tensors F lk and H lf According to
(IV. 80, 80') the tensor F lk defines a pair of space vectors B and E in the
arbitrary system of coordinates >S r by the equations
B (#>i, /'ai,/^)' ?E (*\H ^42^43) (^)
B is an axial vector and E a polar vector Similarly, the tensor H lk
defines a polar vector D and an axial vector H by the equations
H (// M , //,//,.,), *D (//,//,//) (34)
Further, consider a four-vector with the components
/, (J/c,V>) (35)
in the system *S f When the components of the tensors F lk , 7/, A , and J t
are given m one system of coordinates, we can calculate the components
of these tensors in any other system by means of the transformation
equations (IV SI') and (IV 29) of antisymmetncal tensors and vectors.
For the components of J L we thus get
v (v
(36)
\ V / /
where v is the velocity of 8 f relative to S.
VII, 73 ELECTRODYNAMICS, THERMODYNAMICS 197
If we now define the tensors F lk , 7/, A , J l so that the quantities E, D,
H, B, J, p are identical with the macroscopic electromagnetic variables
E, D, H, B, J, P Q in the rest system of the matter, the field equa-
tions of electrodynamics in any system of coordinates must have the form
(37 a)
- ' <
For in the rest system S the equations (37) are then identical with
Maxwell's equations (31 ) and, since they are tensor equations, they must
hold in any system of meitia On account of (IV 187) the equations
(37 a) may also be wiitten ,,-n*
'*=-<), (37 a')
^A
where F*^ is the pseudo-tensor dual to F, k
Using (33), (34), (35) in (37) we thus get in e\ er\ system oi co-
ordinates . , p*
curlE-f- =- 0, di\ B 0, (38 a)
c f't
curl H - l fD -= J , div D - - p (38 b)
c dt c
The quantities J l -- ( J/r, ip) may be interpreted as current and charge
densities in the system /S r , foi from (376) we get the continuity equation
^ ,_ ' 2// ^ (39)
i ]l i ^i'^k
Further, m an insulator, wliere J -- in the rest system, we get fiom
the equations (3(i)
where u is the velocity of the ponderable matter relative to <S* Hence
Se -=/>8r --= /o8F, (41)
i e the charge contained in an infinitesimal piece of matter of volume
87 8F^/(1 u 2 /c z ) is invariant and the current J is a puie conve(;tion
current
In the general case, the current density can always be written as a
sum of the convection current pu and the conduction current C
J-pU + C, (42)
108 NON CLOSED SYSTEMS VII, 73
but this separation is not relativist ically invariant Even if p Q in
the rest system, so that J C is a pure conduction current in this
system, the charge density p in 8 will be different from zero, which
means that we also have a convection current pu = m 8. In fact,
we get from (3ti) in this case
(43)
We can, however, make a relativistically invariant decomposition of J t
by writing QJJ
P '
c,
where p () is the (invariant) charge density in the icst system 8 Q of the
matter, and , , *
U I u 9 u I
is the four- velocity of the ponderable inattei s l (s,.s' 4 ) is a four-
vector with the components
- (JV.O) (45)
in AS' Thus
While,/, has a diiect physical meaning, the field variables E, D ? H, B,
occurring in the field equations (38) have no simple physical significance
m contrast to the field variables E, D, H, B in the rest system,
which could be detei mined by simple macroscopic experiments So far
they are defined only by the transformation equations by which they
may be expressed in terms of the field variables in the rest system 8.
Let us now consider the four-vector F t defined by
From (33) we get for the components of F t in 8
V ( E ~ l (^Ku^B) j(E.u)/r_
'" " aa ' a
Its components in the rest system are thus
F? --- (E,0), (49)
VII, 73 ELECTRODYNAMICS, THERMODYNAMICS 199
i e F l is the four-force acting on a unit charge placed at rest relative to
the matter in a longitudinal crevasse cut in the matter If we put
-(uxB), (50)
c
(48) may be written
and a comparison with (IV. 54) shows that is the force on the test body
in question measured in the system S.
Similarly, the four- vector
K .= l n t jJ } = / D _ +(1 /l ( - u . xH .), 1^
U)/C 1 (52)
with the components K ~ (D, 0) in the rest system, represents the
four-force on a unit test body mseited at rest in a transverse crevasse
Thus also
D-D+^uxH) (53)
r
has a simple physical meaning . it represents the force on the test body
in question measured in the system 8.
Further, if F* k and H* k are the pseudo-tensors dual to F lk and H lk
(see IV 108, 109), we can form the two pseudo-vectors
F* = - 1 n u k = f B -^-"- E) l, j^.'D/f.
(B-)/c . (54)
K * - fi* u ~
1 ~ c lk k ~
(55)
^* and ^Cf are obviously the four-forces acting on a unit magnetic pole
placed at rest relative to the matter in transversal and longitudinal
200 NON-CLOSED SYSTEMS VII, 73
crevasses, respectively, as is seen at once when one considers the com-
ponents of these pseudo- vectors in the rest system. Thus
6=,B_ U **, (56)
fl^H_ (57)
C
are the forces on these unit magnetic poles measured in the system 8.
The vectors , D, fi, B (or the four-vectors F t , K v F*, K*) may thus
in principle be obtained by direct physical measurements performed
by an observer in $. By means of (50), (53), (56), and (57) we can now
also express F tk and H lk m terms of the quantities F v K L , F*, K*. We get
(58)
where 8, A/m is the Levi- ( 1 ivita symbol defined in 43 The equations (58)
are easily seen to be true in the rest system 8 and, since both sides of the
equations transform like tensors, they must hold generally
Since the vector u is a constant we have
curl(uxB) - - (ugrad)B-f udivB -= (ugracl)B,
curl(uxD) - -(ugrad)D-fudivD - - (ugiad)D+pu
Therefore the field equations (38) can also be written in the form
curl B -f ' d * - 0, curl fi - l ^ , . C/r, (59)
c at c at
divB -- 0, divD = p, (60)
i </B <^B , .. dD dD . ]XT .
where ^ _- ^ -(-(ugiad)B, d( - +(ugrad)D
are the substantial time derivatives of B and D, and C is the con-
duction current defined by (42)
So far we have consideied only one material substance moving with
constant velocity u However, since the field equations are linear, the
fields are additive and the equations (37) must hold also in the case of
several bodies, separated by a vacuum, moving uniformly with different
velocities The field equations (37) will, however, lepresent a good
approximation for a system of moving bodies only as long as the accelera-
tions of the bodies due to the electromagnetic forces can be regarded as
small.
VII, 74 ELECTRODYNAMICS, THERMODYNAMICS 201
74. The constitutive equations in four-dimensional language.
Boundary conditions
According to the first t\vo equations of the set (32) the forces on a unit
test body placed in transversal and longitudinal crevasses are propoi-
tional, the constants of proportionality being and ju, according as the
test body is an electric or a magnetic pole Therefore we must have
T) - *, 8 = jufi (6i)
or K^eF^ Ft-f^Kf. (62)
These equations may also be written
tf.^-^*^. (63 a)
Ff k U k = nH?,l\, (63 ft)
the last equation being identical with the tensor equation
F.^+FulM.r^Mil'.+ HuUt + I/.il'J (63r)
In the rest system equations (61), (62) or (63) reduce to the first two
equations of (32) The last equation of the set (32), Ohm's law, may be
put into the form ^ ^^ (64)
as is seen from (45) and (49) when the vector equation (64) is written down
in the rest system Since s ( i \ F t L \ -~ 0, only the first three equations
(64) aie independent, i e (64) is equivalent to
s = "^ (6.-,)
v(i- 2 A- 2 )
On account of (46) and (47) Ohm's law can also be written
J. + ^I'^^W*- ()
The field equations (37) together with the constitutive equations (63)
and (66) enable us to deteimme the field when the charge and current
distributions are known
At the boundary between the ponderable matter and the vacuum the
tangential components of and fl must be continuous, as is seen in
the usual way from (51)) by integrating these equations over infinitesimal
surfaces bounded by a small rectangle with two opposite sides imme-
diately inside and outside the boundary of the matter It is here
understood that the u occurring in the definition of and fl is put equal
to the velocity of the matter also outside the boundary Further, we
find by integration of (60) over a small cylinder with end surfaces
immediately outside and inside the boundary that the normal component
202 NOX-('L()SKI> S\STKMS VII, 74
of B must be continuous at the boundary, while the change AZ) n in the
normal component of D is equal to the surface density of charge on the
boundary
75. Electromagnetic energy -momentum tensor and four-force
density
In Chapter V we have seen that the four-force density in the electron
theory is given by f
Ji " AA, S A.'
where s t is the current density of this theory. This expression followed
irn mediately from the observation that by the very definition of the electric
field strength the force density in the rest system of the charge is yoE. In
ponderable matter with e and ^ different from 1 it is not so easy to find a
unique expression for the force density acting on the matter In the first
place, we have in general a conduction current in the rest system of the
matter, and even in an insulator it is not evident that the force in the rest
system is pE because the elect i ic field strength is defined as the force on a
unit test body placed in a cre\ asse cut in the matter This uncei tainty in
the definition of the force density gives rise to a corresponding uncertainty
in the definition of the electromagnetic energy-momentum tensor.
However, let us consider the four-vector F ll J l which is the analogue
of the four-force density in the electron theory From the field equations
(37) we get
f)// /A ;>(^// /A ) M\i n
T ,f'Ji -- I 1 .] - -- flii
k
V It
Hence F tl J t V F-H U =- -, (67)
^A
S,*- ( 68 )
From (33) and (34) we get for the components of this tensor
(69)
where l iK - ,KJ) K ^H L /^-|(E D+H.B) B llt
in the rest system is identical with Maxwell's stress tensor in ponderable
matter.
VII, 75 ELECTRODYNAMICS, THERMODYNAMICS 203
Further
(*V^42>^43) -= -S I
where S-r(ExH) j
is Poyntmg's vector, and
#44 -=- &, h -- KE.D + H B). (71)
In the rest system S and h are identical with the usually recognized
expressions for the electromagnetic energy flux and energy density in
stationary matter
1 (72)
where g =-. (D x B)
c J
The form of the equations (07) suggests that the left-hand side of (67)
is the electromagnetic four-force density f l and that *V tA as given by (68)
represents the electromagnetic energy-momentum tensor This would
mean that the quantities S, h, g in e\ cry system of coordinates should be
mteipreted as the electromagnetic energy flux, energy density, and
momentum density, respectively The above expressions for S, /?, and
g are due to Mmkowski,*)* in the case \\hen e -- ^ I they reduce to
the corresponding expressions of the electron theory.
If we confine ourselves to homogeneous and isotropic bodies it is easily
seen that the second term on the left-hand side of (67) is zero In the rest
system this term is
on account of (32). Thus, if e and p. arc constant, this term is zero in S ,
but a vector with zero components in one system of coordinates is zero
in all systems Hence, inside a homogeneous and isotropic body we have
(73)
ie f =-/>E-f T (JxB) -p(E + -(uxB)\ (---(CxB))
C C (74)
-(E J) = -E(pu+C)- -[f.ufE C]
C C C
t See ref p 195
204 NON-CLOSED SYSTEMS VII, 75
In the rest system this expression for/ 4 is in accordance with Joule's
expression for the heat q developed in the body per unit time and volume.
In fact, we have
' 4 c ' c c
Iii an arbitrary system of coordinates (f . u) is the mechanical work, and
q ( C) thus must represent the heat production in accordance with
(IV 217) The four equations
/. = -* < 76 )
represent in the usual way the momentum arid energy laws
From (73) we get
Accoidmg to the consideiations m 50, we are here dealing with a typical
example of forces which pioduce a change in the total proper mass of
the matter
Minkowski's elect lomagnetiQ energy-momentum tensor satisfies the
Ham identity ,V,, -*;,// (F U H U ) ^ (78)
as the tensor of the electron theory, but it is not symmetrical, i e
S.A /A, (79)
In the rest system iS' the space part (09) is symmetric for an isotropic
body on account of (32), but for the mixed space-time component we
ha\ e in >S
#?4 - -^c - *c((/? - ^) = i(ep,- I )(E X H) t ^ (80)
In any other system of leference we therefore also have*S l/f ^ S KL even in
an isotropic body
This non-symmetry of Minkowski's energy-momentum tensor has
given rise to a long discussion in the literature | It was generally felt that
this property represented a real difficulty for Minkowski's theory
Abraham J theiefore tried to construct a symmetrical expression for the
electromagnetic energy-momentum tensor In the rest system $
Abraham's tensor agrees with (69), (70), and (71) at least for isotropic
t M Abmham, Itend Pal 28 (1909) , Ann d Phtj* 44, 337 (1914), \V Dalleiibach,
o]) i-it (p l ( )a), M \on Lauo, Die Kelatu'itaMheone, vol t, 3rd ocl , ^ 24, Biaunschxieig,
HU9, \V Pauli, EmyU d Math II ?,s,s vol 2 (1920), p M7,R C Tolman, Kelatinty,
Tfnimodynanncs, and ("ostnoloyy, 54, Oxford, 1934, Jg Tainni, Journ oj Phy*
I'SXR, 1, 439 (1939)
| M Abraham, Rend Pal 28 (1909), Abraham Becker, Theorie der Elcktnzitat,
\ol 11, 6th od , Leipzig, 1933
VII, 75 ELECTRODYNAMICS, THERMODYNAMICS 205
bodies, but instead of (72) Abraham assumed the electromagnetic
momentum density to be given by
g= 1 (ExH)^ S - (81)
c c*
in the rest system Since Abraham's tensor $^? ir is symmetrical in S
it is symmetrical m any system S, but in any system other than S the
components of 8ff r are not simply given by (69), (70), (71), and (81),
the expression of $^ hl in terms of the field variables containing also
the velocity u of the matter in a complicated way The four-force density
/^ br derived from this tensor by the equation
( *<A
also deviates in general from (73) in a complicated way In the rest
system we obviously have
fAbr _ f + f/f_- 1 ^ S 5 yM>i __ y^ (83)
i e in the rest system Abraham 's force density differs from Minkowski's
expression by the term ^ o This term is generally so small that
an experimental verification would be very difficult
Until quite recently, most physicists were inclined to adopt Abraham's
theory However, the question was never quite settled, and recently
Tammf has taken up the discussion again and he comes to the conclusion
that Minkowski's expression for the energy-momentum tensor is correct.
In the first place it should be remarked that, since an electromagnetic
field in ponderable matter is an essentially non-closed system, there is
no a priori reason for the symmetry of the energy-momentum tensor.
Abraham 's main argument for a symmetrical tensor was that the quanti-
ties of the macroscopic theory must be derivable from the corresponding
quantities in the electron theory by averaging over appropriate space-
time regions and, since the microscopic energy-momentum tensor s }k
is symmetrical, the averaged tensor ,v lfr must also be symmetrical But,
as remarked by Tarnm,|. the macroscopic tensor S lk is not simply the
average of s lk , ti lk must rather be defined so as to give the correct force
density and the correct moment of force, i e. we must have
' dx k dx k
t Jg Tamm, see ref., p 204
| Idem, private communication
(84)
206 NON-CLOSED SYSTEMS VII, 75
and, on account of (10),
>,, -, - _ . _.
From (84) we can only conclude that
#i* : - ^~-KA>
where tA may possibly he a non-symmetrical tensor satisfying
**'* = 0.
ar A
Fiom (85) we see that ti lk will be symmetrical only if
l = _*, ^+ r A ^' (86)
' A V 7
and this is not necessarily the case
Further, Tamm could show that Abraham's expression in some special
cases leads to wrong results, while Minkowski's expression for the
energy-momentum tensor is m accordance with the electron theory. It
must also be mentioned that Dallenbach| fiom perhaps not quite cogent
arguments lias given a general derivation of Mmkowski's tensor from
the electron theory I n the following section we shall meet another strong
argument in favour of Minkow ski's theory
While thus the electromagnetic energy-momentum tensor is non-
sym metrical, we may assume that the total energy-momentum tensor of
matter and field is symmetric, since we then have to deal with a closed
system. This means, however, that the mechanical energy -momentum
tensor of the matter must also be non-symmetrical This is not in contra-
diction with the considerations in 65, since we there considered a closed
mechanical system, and the expression (VI 66) for the momentum
density was derived from the explicit assumption g S/c 2 , which must
be abandoned in our case if we adopt Minkowski's expression for the
electromagnetic momentum density
76. The propagation velocity of the energy of a light wave in a
moving refractive body
In Chapters [ and 11 we have defined the direction and velocity of a
light ray in a transparent refractive body by means of Huyghens's
principle, and in 24 it was shown that the ray velocity so defined trans-
forms like the velocity of a particle by Lorentz transformations, i e by
the equations (II 45-47 ) As a consequence of these equations we arrived
t See ref , p 195.
VII, 76 ELECTRODYNAMICS, THERMODYNAMICS 207
in 25 at the aberration formula (II 91 ) and at Fresnel 's formula (II 92)
which, as regards effects of the first order, are in agreement with the
experiments.
According to Maxwell's theory of light, optical phenomena in a re-
fractive body with the refractive index n are described by means of
Maxwell's phenomenological equations of electrodynamics for a sub-
stance with the electric and magnetic constants and /u, connected with n
bv the equation .
=->/M ( 87 )
At least this is true for sufficiently long waves, where we can neglect all
dispersion phenomena Further, a transparent body which does not
absorb any light must be regaided as a perfect insulator, i e we have
a -- 0, J - 0, p ~ (88)
Now the ray velocity must be identical \vith the velocity with which
the energy in the wave is propagated In the aberration expenment, for
instance, the angle ot aboi ration is the angle through which the telescope
must be tilted in order to get the ray, i e the eneigy, into the telescope
The direction of the ray velocity must therefore be the Scime as the direc-
tion ot propagation of the; energy in the wave When the energy-
momentum tensor of the electromagnetic' field is given, we can, however,
iind the velocity ot the energy by means of (VJ 9), i e
u* Sfh (89)
We must therefore require that u* in the case of a light wave transforms
as a particle velocity by Lorentz tiansformations. This means that the
quantity (VI 15) must be a four-vector As shown m 62, this is the
case only if the energy-momentum tensor satisfies the condition (VI 19).
We shall now show that this condition is actually satisfied by Minkowski's
tensor (68)- (72), but not by Abraham's tensor, and this is a strong
argument in favour of Minkowski's theory
As mentioned in 62, it is sufficient to prove the validity of the equa-
tion (VI 1 9) in one system We choose to work in the rest system of the
refractive substance. We can obviously confine ourselves to the con-
sideration of a plane wave, for in the problems which were considered in
Chapters I and II the rrxlii of curvature of the wave fronts are large com-
pared with the wa\ e-length (geometrical optics) and therefore the curved
wave fronts can at each place be approximated by plane waves,
As shown in Appendix 3, the most general solution of the field
equations with p -_ 0, J 0, representing a plane wave with the wave
208 NON-CLOSED SYSTEMS . VII, 76
normal n in the rest system, is
E _/(<" (x.n)/ W ) ^-|x.n)At')
00)
-. e(1 , ,-
" - " / ' " ^ I / "
V/u, v/x
Here e (1) and e (2) are two fixed unit vectors which are perpendicular to
each other and to the unit vector n, i e
(e (1) .e< 2 >) =- (e (1 >,n) -- (e (2 >.n) =- 0, (e^e,,) -- n (91)
/and g are arbitrary functions of the argument / (x n)/w, and
w = , C ^ C (92)
is the phase velocity
Thus we get from Minkowski's expressions (68) (72) m the rest system
S-r(ExH)^ C f/ 2 +f/ 2 )e, (93)
VM
where e -= (e t X e 2 ) ^- n, (94)
i e the direction of propagation of the energy e coincides with the
direction of the wave normal n in this system Further, we have
h- |(e 2 ^// 2 )-/ 2 H/ 2 (95)
Hence u* = - = ' _e i/*e --= e w, (96)
h
i e the velocity of the energy is equal to the phase velocity in the rest
system From (96) we get, by means of (VI 15 and 12),
S l = (S, ich) = (r f ? a )- e, ic (98)
Further, we get from (69)
2 +f7> t ^, (99)
since ^ 1) d 1) -f'^ ( i 2> 4- g ^K = 8 tK
on account of (91). Finally, (72) gives
*+!/ 2 K. (100)
VII, 76 ELECTRODYNAMICS, THERMODYNAMICS 200
For the quantity
we then get, by means of (97), (99), and (100),
c
Thus the tensor R lk defined in (VI 19) has the following components;
Hence the condition (VI 1 9) is satisfied and the velocity of propagation
of the energy u* is in every system of coordinates identical with the ray
velocity as determined by Huyghens's principle f
If v' is the velocity of the rest system S relative to a system of co-
ordinates *S y/ with the same orientation of the spatial axes as in A>, the
transformation coefficients e (lc in (VI 17' or 18) are given by (IV 120)
with v v', i e
Since the velocity of the energy in $' is
this quantity maybe calculated from (VI. IS or 17'), using (104). We
then get ( , }
u* r =- r e+v' --i } , (106)
<J(JJ.) ^
which, of course, is in accordance with (II. 55) if we put v v' and
u =- {c/^(jji)}e in this latter formula and neglect terms of order higher
than the first in ?/. For the magnitude of u*' we get
(107)
n n
in accordance with the Tresnel formula' (II. 92).
t See also A Scheye, Ann d Phyv (4), 30, 805 (1909)
3595 60 p
210 NON-CLOSED SYSTEMS VII, 76
On the other hand, if we adopt Abraham's expressions for the energy-
momentum tensor the equations (90)-(99) will still be valid in the rest
system, but instead of (100) we get from (81)
i O l C
. A. I . . ' / -w^ f f v ^ * i f O i 9\xv Jt I 1\ / 1 f\ O \
+ <7 ) e ~- & (& i) ,,. (108)
C L
Hence, for the quantity (101) we get in this case
and the tensor K lk will not be zero any more Thus l r * will not be a
four-vector and u* will not transform like a paiticle velocity.
To find explicit expressions for the \ector u*' we again use (104),
(105), and (VI 17'). Hence, with Abraham's expression for the energy-
momentum tensor we get for the velocity of the electromagnetic energy
in the system >S"
e+v'- (v '- e)c + (v' e)l- Me, (109)
which deviates from the ray velocity as defined by rluyghens's principle
by the last term Instead of (107) we now have
(109')
From (109) we see that the direction of the energy flux in this case is
different from the direction of the ray velocity as defined by Huyghens\s
principle, which would give rise to a change in the aberration formula
for light traversing a medium of refractive index n - 1, the deviation
being of first order in r' . Unfortunately it is very difficult to measure
even first-order aberration effects in moving transparent media
Further, we see fiom (109) that the velocity of the energy differs from
the phase velocity even if v' and e are parallel, in contrast to the ray
velocity which in this case is identical with the phase velocity This
strange result is connected with the following circumstance
While Mmkowski's four-force density/, is zero in the case considered,
Abraham's theory would give a non-vanishing force density on a homo-
geneous insulator In the rest system we have, according to (83),
> f = . (HO)
dt
VII, 76 ELECTRODYNAMICS. THE KM OD\ N \M ITS 211
Thus, in this system, the electromagnetic 1 energy is conserved, but this
will not be the case in /$" From (104) and the transformation equations
f'Abr __ /Abr i fAbr
Ji -Ji ~T- ikJk
we get
(111)
which is i/c times the mechanical work on the substance per unit time
and volume Thus, in *S' we have an exchange of energy between the
electromagnetic and the mechanical system, i e a local absorption and
re-emission of light energy by the body This clearly shows that Min-
kowski's decomposition of the total energy-momentum tensor into an
electromagnetic and a mechanical part is more natural than Abraham's,
a transparent body being in M in ko \vski\s theory a system which does
not even locally exchange energy with the electromagnetic field.
77. The laws of thermodynamics in stationary matter
As shown by Planckf and Einstein;]., the usual laws of thermodynamics
may be easily incoiporated in the special theory of relathity For
simplicity w r e shall confine ourselves to the consideration of systems
consisting of a thermodynaimc Hind which can exert a normal pressure
only on any sin face element In the rest system of the fluid the two laws
of thermodynamics may be stated in the usual way
According to the first law r , the total energy of the system is a unique
function of its state In a thermodynamical process which gives rise to a
change of state, the change in energy d K Q is given by
dE Q -- S^-fS/n (112)
where 8Q is the amount of heat transferred to the system by the process,
while 8.4 is the mechanical work done by the surroundings on the
system In a reversible process in which the volume V of the system
is increased by an infinitesimal amount, we have
-p Q dV ( \ (113)
where p is the pressure
Accoiding to the second law of thermodynamics, the entropy 8 in
the rest system is a function of the thermodynamical state The change
t M Planck, Berl fier , p 542 (1907), Ann d. Fliyt 76, 1 (1908), F Hasonohrl,
WienBer 116, 1391 (1907)
} A Einstein, Jafnb f Had nnd El 4,411 (1907)
212 NON-CLOSED SYSTEMS VII, 77
of entropy content of a system by a small change of state is defined by
where SC/^v is the amount of heat transferred to the system in a reversible
process winch brings about the change of state considered, and T is the
absolute temperature of the system. On the other hand, if the change of
state is brought about by an irreversible process, we have always
78. Transformation properties of the thermodynamical quan-
tities
We shall now establish the transformation properties of the thermo-
dynamical vai lables. Consider a system of inertia with respect to which
our thermodynamical system is moving with the constant velocity u.
Since the fluid in question is in equilibrium in the rest system, the total
momentum and energy are given by (VI 1 1 2, 1 13) which together with
(VI 100) and (II 34) lead to the equations
G _ w - va)
,-J/l --,r-> Jfl-</r,<l ' (|ig)
Now consider again a reversible process which brings about a change
in the state of the system In an arbitrary system of coordinates the
first law of thermodynamics may then again be written
dK - 8Q + &A, (117)
where &Q and 8^4 are respectively the heat How through the boundary
and the work done by the surroundings on the system by the process
considered. This work is, however, not simply equal to p dV, as it was
in the rest system. In order to preserve the equilibrium of the substance,
i.e a constant velocity throughout the body, we must assume the exist-
ence of external forces besides the normal pressure The total extra force
in the body, which is necessary to keep up a constant u in *S\ is
F ... ^ = - 7 - - *(K+iM">), (118)
at c 2 \!(\~~u*/c 2 ) dt
and, since the right-hand side is different from zero during the thermo-
dynamical process considered, there must be an external force on the
VII, 78 ELECTRODYNAMICS, THERMODYNAMICS 213
system which carries out work of amount (F.u)dt ^ (u dG) during
the time dt. For the total work BA on the system we thus get
8^4 -^-/M/r+u.r/G, (119)
with G given by the first equation (116), From (1 17), (1 19), and ( 116) we
thus get for constant u
SQ =-= dE +pdV-u.dG
^ dE \ (u*/c*) d( P ^)~(u*/c*)(d^
^!(i-~n 2 /c 2 )
= (dE-)rp dV Q )^(lu 2 /c 2 ). (120)
Hence, by means of (112) and (113),
SQ =~- SQ *J(l~u 2 /c 2 ). (121)
This transformation formula for the amount of heat transferred
through the boundary is identical with the formula (IV. 66) lor the non-
mechanical energy produced inside a body, for instance in the case ot
the Joule heating effect
Further, we define the entropy and absolute temperature in any system
of coordinates by the equation
dS = 8 Q-. (122)
Thus, for any reversible process without heat transfer, the entropy is
constant. Consider now a thermodynamical system in some internal
state originally at rest in a definite system of coordinates. If this system
is accelerated reversibly and adiabatically to the velocity u without
any change in its internal state, the entropy must be constant during
the process on account of (122). Thus the entropy of the system is
independent of the velocity when the internal state is the same, which
means that the entropy must have the same value in every system of
coordinates (cf. the analogous consideration, p. 141), i.e. 8 is an
,S'^,S">. (123)
From (122), (121), and (123) we then get the following transformation
formula for the temperature
T - T<V(l-~tt 2 A l2 ) ( 124 )
For any irreversible change of state we have in every system of co-
ordinates ~ n
dS > 6 . (125)
214 NON-CLOSED SYSTEMS VII, 79
79. Four -dimensional formulation of the laws of thermo-
dynamics
In 69 it was shown that the laws of conservation of energy and
momentum in differential form could be expressed by the tensor equation
PiT
-' " 26)
This equation comprises the first law of thermodynamics. It is now
possible also to express the second law of thermodynamics in four-
dimensional language, f
Since the entropy is an additive quantity we can introduce the
entropy density s defined so that 8$ = s 8V is the entropy content of
the volume element 8V. From (123) and the last equation (116) we get
at once the transformation equation
s - " 27)
for the entropy density. For the change of entropy in the infinitesimal
time 8t of the element considered we then have, according to (125),
>, (128)
where 8Q is the heat transferred to the element in the time 8t Here,
d(s 8V) /dt is the substantial time derivative and, thus, we get by means
of (IV. 200 and 204)
= g+(grad) S+S divu
= (div( SU ) + ~)sr. (129)
We can now introduce the four-current density of entropy
(130)
\ C
by analogy with the four-current density of electric charge (V 3) On
account of (127) and (IV. 39) the four- vector /S, may also be written
(131)
c
by analogy with (V. 9).
f K C. Tolman, Rclatn'ity, Thcrmotli/ttamiif and Cosmology, 71, Oxford 1934.
VII, 79 ELECTRODYNAMICS, THERMODYNAMICS 215
By means of (129) and (130) the left-hand side of (128) takes the form
c 'SKSJ = -*8S, (132)
d,r t 3x l
where Si] 8r8,r 4 /i is the four-dimensional volume element defined by
(IV. 121)
Since the right-hand side of ( 128) is an invariant on account of (121)
and (124), the second law of thermodynamics may be written in the
simple invariant form ,, ^ nQ
*> (133 >
80. Ideal monatomic gases
For a monatomic gas consisting of N molecules we have in the rest
system the usual expressions for the energy, entropy, etc., at least for
moderate temperatures, where the mean kinetic energy of the molecules
is small compared with the rest energy m c 2 of a molecule. Thus if
1, (134)
m c
we have in the rest system the equation of state
(135)
where k is Boltzmann's constant. Further,
#0 - Nn^+lNkT*, (136)
where the energy has been normalized so that E Q is equal to the sum of
the rest energies of the molecules at zero temperature.
From (114), (135), and (136) we get in the usual way
(137)
and by integration
|A T Hn T"+Nkln F + C, (138)
where C is a constant which is independent of T and F.
In a system of coordinates in which the gas has the constant macro-
scopic velocity u, we get by means of (116), (124), and (135)
P V = NkT, (139)
i e. the equation of state of an ideal gas is invariant in form. Further,
N O ___ M
(140)
* S
fi - *NklnT+NklnV-tNlslnJ(l-'U*lc*) + C )
216 NON-CLOSED SYSTEMS VII, 80
The energy-momentum tensor is given by (VI. 104)
(141)
where n = N/V is the number of molecules per unit volume in the rest
system. Thus we have the following relation between the pressure p,
the rest density //,, and the temperature T Q
_ v?kT Q (142)
P -^~+pl"/c'- ( ^
81. Black -body radiation
The electromagnetic radiation inside a hollow enclosure in equilibrium
with the walls at a definite temperature can be treated as a perfect fluid.
In the rest system of the walls the flux of electromagnetic radiation is
zero at every point and, according to 8tefan-Boltzmann 's law, the energy
density A is given by the formula
where a = 7-6237 x 10~ 15 erg cm.- 3 deg.~ 4 (144)
is Stefan-Boltzmann 'a constant. Thus h is independent of the volume
of the enclosure. The radiation exerts a normal pressure p which is
a 7/ i/r 2
P = l^ = - 3 -= M 3 1 - < I45 >
For the total energy of the radiation we now get
(146)
and, on account of (114),
iV Q . * (147)
By integration we get S *aFT 3 , (148)
where the constant of integration has been chosen so as to make 8 =
for F -= and T = 0.
In the system of coordinates, where the rest system is moving with
velocity u, we then get, on account of (116), (123), (124),
4
G -
""
4 /7F7 73
.Q _ * QV
""
. (149)
VII, 81 ELECTRODYNAMICS, THERMODYNAMICS 217
The relation between G, E, , and u is the same as for a spherically
symmetric electrostatic system (cf. (23)).
The energy-momentum tensor for black-body radiation is again
** ( 150 >
with the relation (145) between the pressure and the density of the
rest mass. It satisfies the same relation
as the electromagnetic energy-momentum tensor of an arbitrary electro-
magnetic field (cf. V. 108). This should also be expected, since the macro-
scopic energy-momentum tensor of the black-body radiation is the
statistical average of the electromagnetic energy-momentum tensor in
the canonical ensemble corresponding to the temperature T '.
VIII
THE FOUNDATIONS OF THE GENERAL
THEORY OF RELATIVITY
82, The general principle of relativity
ACCORDING to the special principle of relativity winch is the basis of the
special theory of relativity, all systems of inertia, i.e. all rigid systems of
reference moving with constant velocity relath e to the fixed stars, are
completely equivalent as regards our description of nature. Mathe-
matically this principle found its expression in the covariance of the
fundamental equations of physics under Lorentz transformations In
spite of the inner consistency and harmony which characterize the special
theory of relativity, it is, however, extremely unsatisfactory that this
theory also distinguishes certain systems of reference, the systems of
inertia, from all other conceivable systems of reference. This defect was
felt especially serious in the treatment of the so-called clock paiadox
mentioned in 20 There we actually had to refrain from giving a real
solution of the clock paradox The question was simply rejected by a
reference to the fact that the system of coordinates $*, following the
moving clock, is a system of inertia for only part of the time and that a
discussion of the problem in this system of coordinates therefore falls
beyond the scope of the special theory of relativity.
However, it seems to be rather difficult beforehand to acknowledge
the accelerated systems of reference as equivalent to the systems of
inertia as regards the description of natural phenomena (When, in the
following chapters, we speak of an accelerated system, simply, we
always have in mind a system accelerated relative to the systems of
inertia or to the fixed stars ) For example, if we consider a purely
mechanical system consisting of a number of material particles acted
upon by given forces, and with velocities small compared with the
velocity of light relative to a system of inertia, Newton's fundamental
equations of mechanics may be applied with good approximation in the
description of this system. On the other hand, if we wish to describe the
system in an accelerated system of reference, we must introduce, as is
well known, so-called fictitious forces (centrifugal forces, Coriolis forces,
etc.) which have no connexion whatever with the physical properties of
the mechanical system itself. In fact, they depend exclusively on the
acceleration of the system of reference introduced relative to the systems
of inertia.
VIII, 82 THE GENERAL THEORY OF RELATIVITY 219
It was just for this reason that Newton introduced the concept of
absolute space which should represent the system of reference where
the laws of nature assume the simplest and most natural form. However,
as mentioned at the beginning of Chapter II, the notion of absolute space
lost its physical meaning as soon as the special principle of relativity was
generally accepted, for as a consequence of this principle it became
impossible by any experiment to decide which system of inertia had to
be regarded as the absolute system Therefore, Einsteinf advocated
a new interpretation of the fictitious forces in accelerated systems of
reference instead of regarding them as an expression of a difference
in principle between the fundamental equations in uniformly moving
and accelerated systems he considered both kinds of systems of refer-
ence to be completely equivalent as regards the form of the funda-
mental equations , and the 'fictitious ' forces were treated as real forces on
the same footing as any other force of nature. The reason for the occur-
rence in accelerated systems of reference of such peculiar forces should,
according to this new idea, be sought in the circumstance that the distant
masses of the fixed stars are accelerated relative to these systems of
reference. The 'fictitious forces' are thus treated as a kind of gravita-
tional force, the acceleration of the distant masses causing a 'field of
gravitation' in the system of reference considered.
The idea that the acceleration of the distant masses can produce a
gravitational field which is not perceptible in a system of inertia is not
more artificial than, for example, the fact that an electrostatic system
has zero magnetic field in the rest system of inertia of the charges, while
a magnetic field is present in every system of inertia in which the charges
are moving with constant velocity. The cause of the appearance of a
magnetic field in the 'moving' system of inertia must be sought in the
motion of the electric charges relative to these systems, and the appear-
ance of the magnetic field can at any rate not be taken as an indication
that the fundamental equations of electromagnetics have different
forms in different systems of inertia The only essential difference
between the two cases considered is the circumstance that the cause of
the magnetic field can be found in the state of motion of terrestrial
systems (viz. that of the charges), while the origin of the gravitational
fields in accelerated systems must be sought in the state of motion of the
distant celestial masses. Previously the effect of the celestial masses
had been considered to be negligible ; now, however, we must include
t A Einstein, Jahrb f. Rad und EL 4, 411 (1907), Ann d. Phys 35, 898 (1911),
ibid. 38, 355, 443 (1912), Phys ZS 14, 1249 (1913)
220 THE FOUNDATIONS OF VIII, 82
the distant masses in the physical system considered. Only when we
work in special systems of reference, viz. systems of inertia, is it not
necessary to include the distant masses in our considerations, and this is
the only point which distinguishes the systems of inertia from other
systems of reference. It can, however, be assumed that all systems of refer-
ence are equivalent with respect to the formulation of the fundamental laws
of physics. This is the so-called general principle of relativity.
83. The principle of equivalence
The interpretation of the 'fictitious forces' as gravitational forces is
corroborated decisively by the fact that they have an essential property
in common with the usual gravitational fields, viz. the property to give
all free particles the same acceleration irrespective of the mass of the
particles It is immediately clear that the 'fictitious forces' have this
property, and Galileo was the first to prove this property for the gravita-
tional field ol the earth As a result of his experiments he was able to
make the statement that all bodies 'are falling with equal speed' in empty
space. This result simply expresses the fact that the force with which the
gravitational field of the earth affects a particle is proportional to the
inertial mass of the particle which determines the inertia of the particle
against changes of motion As long as the velocity of the particle is
small relative to the velocity of light, its motion in the direction of the
gravitational field is therefore given by the equation
mx = wg,
where m is the mass of the particle and x is the acceleration of the particle
in the direction of the gravitational field The quantity g is a measure
of the strength of the gravitational field and is independent of the mass
of the particle This circumstance is frequently expressed by stating
that the ratio between the inertial mass of a particle and its gravitational
mass is a universal constant, depending only on the units in which the
quantities in question are measured. This theorem has now been proved
by a large number of experiments,! the most accurate of which are those
performed by Eotvos and by Zeeman. The ratio of the inertial and
gravitational mass was always found to be the same. A particular
interest was attached to the experiments of Southerns and Zeeman
with uranium, which at that time was known to have a great mass
f R v Eotvos, Math u nalurw Ber aua Ungarn, 8, 65 (1890), Ann. d Phys. 59,
354 ( 181)6), L. Southerns, I' roc ftoi/ tfor London, A, 84, 325 (1910); P. Zeeman, Proc.
Amst 20, 542 (1917) H. v Eotvos, D. PekAr, and E. Fekote, Ann. d Phys 68, 11
(1922).
VIII, 83 THE GENERAL THEORY OF RELATIVITY 221
defect. In Chapter III we have seen that any energy of amount E
corresponds to an inertial mass m = E/c 2 , a theorem which has been
verified experimentally by numerous nuclear transformation processes
(cf. 32). The mass which is determined in a mass spectrograph obviously
is the inertial mass, and Zeeman's result now shows that the binding
energy of the uranium nucleus which appears in the mass defect also
corresponds to a gravitational mass which has the same universal ratio
to the inertial mass as for all other types of mass.
In view of the property just discussed, a gravitational field may thus
be characterized by the 'gravitational acceleration ' independent of the
mass of the test particle, and this applies both to the usual gravitational
fields due, for example, to the gravitation of the earth or the sun, and
to those gravitational fields which appear in accelerated systems of
reference and which are due to the distant masses of fixed stars Actually
the gravitational field on the surface of the rotating earth is a mixture
of these two types of field, the centrifugal force due to the rotation of
the earth being in general* not negligible compared with the force due
to the attraction of the test body by the mass of the earth. It is thus
quite natural to assume that both types of gravitational fields are of the
same nature and obey the same fundamental laws. This assumption is
often referred to as the principle of equivalence. It is true that the
gravitational fields due to the distant masses can be made to disappear
by a suitable choice of the system of reference, viz. by choosing a system
of inertia as system of reference, while the gravitational fields arising
from 'close' masses such as that of the earth or the sun cannot be
'transformed away' by a proper choice of the system of reference; the
latter fields will therefore be referred to as permanent gravitational fields.
In this respect, however, the situation is quite similar to the case of
the magnetic fields with which the gravitational fields were compared
in 82. In some cases, viz. when the charges producing the electro-
magnetic field have the same constant velocity relative to the fixed stars,
it is possible completely to transform away the magnetic field by choosing
the rest system of the charges as system of reference, for in this system
the field will be a purely electrostatic field. In general it will, however,
not be possible to choose a system of reference in which the magnetic
field disappears everywhere in this system. Nevertheless, in this case
the electromagnetic field is not considered as essentially different from
the field of a system where the magnetic field can be transformed away.
In all cases the electromagnetic field obeys the same fundamental equa-
tions, viz. Maxwell's equations.
222 THE FOUNDATIONS OF VIII, 83
The most important task will now be to look for the general funda-
mental laws which all types of gravitational fields must obey. In the
first place, we must, however, try to find the field functions which can
give an adequate description of the fields of gravitation. To this end we
shall first consider the simple case where no permanent gravitational
fields are present. By a suitable choice of the system of reference, viz.
in a system of inertia, the field of gravitation will disappear and we can
apply the laws of the special theory in this system A simple transforma-
tion to accelerated systems of reference permits the determination of the
field quantities describing the gravitational fields in accelerated systems,
and, according to the principle of equivalence, these quantities may be
assumed to give a correct description also of the more general permanent
fields of gravitation.
84, Uniformly rotating systems of coordinates. Space and time
in the general theory of relativity
The development of the ideas underlying the general principle of
relativity now leads, as we shall see, to a still more radical revision of our
conceptions of space and time than that required by the special principle
of relativity . To illustrate the character of the problems with which we
are confronted, we shall start with the consideration of a very simple
accelerated system of coordinates, viz. a rigid uniformly rotating system.
Let / denote a certain system of inertia in a part of space so far from all
masses that all gravitational effects can be neglected; further, let X, Y ',
Z, T denote the usual space-time coordinates defined in the way discussed
in 16 and 17. Instead of Cartesian coordinates we can obviously just
as well employ general curvilinear coordinates for the fixation of the
points in physical space. If we confine ourselves to the consideration of
events in the ^YF-plane we can, for instance, introduce polar coordinates
(R,9) by means of the equations
x -^ Rcose, y=-j?sin0. (i)
Now we can define a uniformly rotating system of coordinates S
with spatial coordinates
x = rcos#, y =- rsin# (2)
by means of the transformation equations
r = R, # = 6-ajT. (3)
Any fixed point in the rotating system corresponding to constant values
of (x,y) or (r,$) obviously performs a circular motion relative to / with
the constant angular velocity cu. For T = the two systems of co-
ordinates coincide.
VIII, 84 THE GENERAL THEORY OF RELATIVITY 223
Thus for all points with
r - R < c/aj (4)
the rotating system of reference may be represented by a uniformly
rotating material disk. Each point p on this disk is characterized by a
pair of numbers (x,y) or (r,&) which are equal to the coordinates (X, Y)
or (R,B) of that point in the fixed JCF-plaiie with which the point p
coincides at the moment when the clocks in the system of inertia /
record the time T ~ 0.
For the measurement of distances between fixed points on the rotating
disk we shall use standard measuring-rods of the same kind as those used
in the systems of inertia, but now at rest relative to the rotating disk.
In connexion with the process of measuring distances in accelerated
systems of reference a problem arises which did not occur in inertial
systems. If the measuring-rods in one way or other are kept in a fixed
position relative to an accelerated system of reference, they will generally
be submitted to forces which may cause a deformation of the measuring-
rods ; for, according to the special theory of relativity, no absolutely
rigid bodies can exist, since they would provide a means of trans-
mitting signals with velocities larger than c.
Consider, for example, a measuring-rod, one end of which is attached
to the point (/%#) on the rotating disk and which lies in the direction
of the radius The centrifugal forces will then undoubtedly cause a
lengthening of the measuring-rod This deformation will, however,
depend on the elastic properties of the material from which the
measuring-rods are made, and all such deformations of the measuring-
rod can therefore easily be corrected for. Now we make the assumption
that measuring-rods on the disk, after insertion of these corrections, have
exactly the same length relative to I as the standard measuring-rod in the
system of inertia 7 which at the moment considered has the same velocity
as the measuring -rod on the rotating disk. In general, we shall assume
that the (corrected) standard measuring -rods in an accelerated system relative
to the measuring-rods in I are subjected to Lorentz contractions only, which
means that the lengths of the rods are independent of the accelerations
relative to 7.
If we therefore measure the distance between two points (r,#) and
(r+dr,#) on the disk with a standard measuring-rod on the disk we get
the value -, -, ,-\
dcr dr; (5)
for the velocity of the measuring-rod relative to 7 is perpendicular to
the direction of the rod and thus does not give rise to any Lorentz
224 THE FOUNDATIONS OF VIII, 84
contraction. On the other hand, if we consider two points with the co-
ordinates (r,&) and (r,&+d&) on the disk, a measuring-rod connecting
these two points will have the velocity ra> relative to / in its own direction
and it will therefore be contracted relative to the measuring-rods in /,
in accordance with Lorentz's formula (II. 33). Hence the distance
between the two points measured with the contracted measuring-rod
For the distance da between two neighbouring points (r,#) and
(r+dr,#-f d#), measured with a standard measuring-rod on the disk,
we get similarly
This follows immediately by means of the Pythagorean theorem if we
consider the measuring procedure from the point of view of an observer
in the system of inertia /, taking into account the fact that the measuring-
rod moving with the rotating disk relative to the measuring-rods in /
is contracted in the direction of the velocity, but unchanged in a direction
perpendicular to the direction of motion.
It is now immediately clear that the geometrical theorems obtained
by means of measurements with standard measuring-rods at rest relative
to the disk in general will deviate from the theorems of Euclidian
geometry. Consider, for example, the curve given by the equation
r constant.
According to (5) this curve will represent a circle with radius r. The
periphery of this circle will, however, according to (6), have the length
27T
rd&
o
Therefore the ratio of the length of the periphery to the radius will not
be 27T, but 2
Consequently we see that the recognition of the general principle of
relativity, according to which the accelerated systems of coordinates
are equivalent to the systems of inertia for the description of nature,
forces us in some cases to abandon the Euclidean geometry which, as
particularly advocated by Kant and even in the special theory of rela-
tivity, was regarded as an indispensable foundation of all description of
space. This also has the consequence (cf. 87) that it is not possible in
VIII, 84 THE CSKN T ERAL THEORY OF RELATIVITY 225
general to use Cartesian space coordinates in accelerated systems of
reference and that we have to make use of general curvilinear coordinates
for the specification of the points in physica4 space.
Similarly, the general principle of relativity also requires a renewed
revision of the notion of time. In the special theory of relativity the
time in a system of inertia was simply defined by means of standard
clocks placed at different points in the system and regulated by means
of light signals m the way described in 16. Two clocks which had been
synchronized in this way remained synchronous. Likewise we could now
think of defining the time in the rotating system of coordinates by means
of standard clocks inserted at rest everywhere in this system and set
according to the standard clocks in the system of inertia /, for example,
by putting them to zero at the moments when the clocks in / with which
they coincide show zero. Then we have for the time t thus defined in the
rotating system t = for T 0. A standard clock at the point r, # on
the disk has, however, a velocity TOJ relative to /, and therefore it will
be retarded relative to the clocks in / in agreement with equation (II. 36),
i e at a later time we have
J = TV(l-r 2 o; 2 /c 2 ). (9)
In accordance with the assumption made in 20, it is implied that only
the velocity, not the accelerations relative to /, will influence the rate of
a standard clock Actually, sufficiently strong accelerations will of
course more or less influence the rate of a real clock (cf. a watch which is
dropped on the floor), but such an effect which depends on the material of
which the clock is made can be corrected for, just as was the case
with the measuring-rods.
The description of time in the rotating system which we obtain by
using the time variable t defined by (9) is, however, although admissible
in principle, highly unpractical. Imagine, for instance, a light source
(an atom) which is placed at the point A with coordinates (r, #) and which
emits light with the proper frequency v The number of waves emitted
in the time-interval from t to t = 1 is then by definition equal to v .
The number of waves emitted during the time from T to T 1 is
therefore, according to (9), v (l r 2 co 2 /c 2 )*. The same number of waves
will also arrive at the centre (r = 0) during the time-interval from
T = to T = 1 or, since t = T for r = 0, in the time-interval from
t = to t 1.
The number of light waves which are emitted from the point A per
unit time in the time-scale t is thus larger than the number of waves
3595.60
226 THE FOUNDATIONS OF VIII, 84
arriving at the centre per unit time, and with this time variable we get
a very complicated description of the propagation of light. This con-
sideration shows that in general it is not convenient in accelerated
systems of reference to use a time variable defined by standard clocks,
and that a much simpler description may be obtained when one uses
clocks of a different rate. In the case of the rotating disk, for example,
it will be most convenient to use coordinate clocks whose rate at any place
is (1 r 2 o> 2 /c 2 )~* times faster than the rate of the corresponding standard
clock, for this means that the time parameter t defined by these co-
ordinate clocks is identical with the time T in 7, i.e. we have the trans-
formation t =- T (10)
instead of (9).
In principle it is admissible, however, to use coordinate clocks of an
arbitrary rate, provided that the time variable t defined by these co-
ordinate clocks gives a reasonable chronological ordering of the physical
events. In accelerated systems of reference the spatial and temporal
coordinates thus lose every physical significance, they simply represent
a certain arbitrary, but unambiguous, numbering of the physical events.
85. Non- Euclidean geometry. The metric tensor
As we have seen, the spatial geometry on the rotating disk is non-
Euclidean. Although all geometrical experience in the three-dimensional
physical space is in complete agreement with the theorems of Euclidean
geometry, the notion of non- Euclidean geometry in two dimensions is
by no means foreign to us, since we meet examples, of such geometries
on every curved surface A well-known example is the spherical geometry
on spherical surfaces. As an introduction to the non-Euclidean geo-
metries in w-dimensional space, we shall therefore consider the geometry
on an arbitrary two-dimensional surface embedded in a three-dimen-
sional Euclidean space. If x, y> z are Cartesian coordinates in this space,
the surface in question may be given by a parametric representation
x - F(x\x^ y - 6V,* 2 ), * - H(x\x*)> (11)
where F, G f , H are given functions of the two parameters x 1 and x 2 in
certain intervals. By differentiation of (11) we obtain
, dF , . , 'dF , 2
dx = dx l -\ -- ax 2
dx 1 ^dx*
7 &H 7 , . dH
dz = - dx l -\ --
1 ^
(11')
VIII, 85 THE GENERAL THEORY OF RELATIVITY 227
The distance ds between two adjacent points on the surface corresponding
to the parameter values (x l ,x 2 ) and (x l +dx l ,x 2 +dx 2 ), respectively, is
iven b ^ ds 2 = dx 2 +dy 2 +dz 2 ,
where dx, dy, dz are linear expressions in dx l and dx 2 given by (11').
Using these expressions we therefore get ds 2 expressed as a homo-
geneous quadratic form in dx 1 and dx 2 , i.e.
ds 2 = g ll (dx^+g ia (dai l da*)+g n (dxVx l )+g^dx^ (12)
with
3F 3F . dG 8G . dH dH
(12 )
The angle between two line elements corresponding to the increments
(S# l ) (Sx l ,$x 2 ) and (A# 7 ) = (Ax 1 , A.r 2 ), respectively, for the para-
meters (x l ,x 2 ) is given by the equation
(13)
,
SsAs
where (So:, Sz/, 8z) and (A#, A?/, Az) are the increments in x, y, z obtained
from (IT) by replacing dx 1 = (dx l ,dx 2 ) by (8x l ) and (Aa: 1 ), respectively,
85 and As being the lengths of the line elements. Applying (11'), (12),
and (12') this equation can be written in the form
cos = l1 i* 2i , 8
The line elements (8x l ) = (Sx l ,Sx 2 ) and (As 1 ) = (Ao; 1 , Ao: 2 ) also define
an infinitesimal parallelogram on the surface with the area
da 85 As sinfl, (14')
where is given by (14).
The curves on the surface which are obtained from (11) by putting
x l = constant, (15 a)
x 2 constant, (156)
respectively, are called coordinate curves. Every point on the surface
is the point of intersection of two coordinate curves from the manifolds
(15a) and (156), respectively. If the line elements (8^') and (A# 1 ') are
lying in the directions of these coordinate curves we have
(&**) - (cfeSO) and (A**) = (
228 THE FOUNDATIONS OF VIII, 85
and we get from (12) and (14)
^ = ten)* ^> AS = (022)* dx* 9
7--77,
(011022)*
where
011 012
021 022
is the determinant corresponding to the scheme of numbers g lk For the
area du of the parallelogram we thus get from (14')
da= <Igdx l dx*. (17)
If the parametric representation (11) of the surface has the property
that every set of values of the parameters x l = (x l ,x 2 ) corresponds to
one and only one point on the surface, x l (.r 1 ,^ 2 ) represents a set of
general curvilinear coordinates (Gaussian coordinates) on the two-
dimensional surface. All the fundamental geometrical quantities can
then be expressed in terms of these coordinates alone without reference
to the variables of the three-dimensional space in which the surface was
supposed to be embedded. lfy tk ~- <J lk (x l ) are given functions of the co-
ordinates (r'), the line element is given by (12),
(Is 2 _-= g lk dx*dx K , (18)
summation over i and k for the values I and 2 being implied in this
expression.
The angle 9 between the two line elements ($x l ) and (Ao; 1 ) is given
by (14) ' o,'Sx^x k
cos 6 y tA . ox LX ___ ^ , .
S -* { }
and the area of the parallelogram defined by two line elements in the
directions of the coordinate curves is given by (17). The quantities g lk
the components of the so-called metric tensor thus determine the
geometry on the surface which in general will be non-Euclidean.
86. Geodesic lines
The straight lines which in Euclidean geometry may be defined as the
curves of shortest distance are in the more general case replaced by the
geodesic lines which likewise may be defined by a variational principle.
Let us consider an arbitrary curve connecting two points P and P 2 on
the two-dimensional surface. In a parametric representation of this
VIII, 86 THE GENERAL THEORY OF RELATIVITY 229
curve the Gaussian coordinates x l Or 1 ,:*: 2 ) may be regarded as certain
functions of an arbitrary parameter A in the interval
A! < A < A 2 ,
i.e. x l - a^A) (i = 1,2). (20)
Let L L(x' l t x l ) be a given function of the variables
x l and x l dx l /d\.
The curve (20) which gives the integral
A 2
f L(x l ,&)d\
A\
a stationary value for all infinitesimal variations of the curve connecting
the fixed points P l and P 2 is then determined by the condition
r*
) L(x\x')d\ ----= (21)
At
for all variations 8^ l (A) satisfying the boundary condition
So^Aj) == So; l (A 2 ) = 0. (22)
Now we have
A 2 A 2
3 | L dX = f Jg 8^(A) + g 8^(A) j ^/A, (23)
AI AI
and since 8x l = d(8x l )/dA, we obtain by partial integration of the last
term in (23), taking into account the boundary condition (22),
This integral can only be zero for all imaginable variations Saf(A) if the
factor in the square brackets is zero along the whole curve. The varia-
tional principle (21), (22) is thus equivalent to the Euler differential
equations
(24)
( ;
If L is a homogeneous function of the variables x l of nth degree, we
have
230 THE FOUNDATIONS OF VIII, 86
and for ft ^ 1 it is easily seen that the function L(x l , x 1 ) must be constant
along the curve defined by the differential equations (24). From (24) and
(25) we get
dL dL ., 3L , d ldL\ ., , dL dx l d IdL . \ dL
___ /yl I /yt ___ I I >>* I __ I fy't I - tn
(26)
Hence (n 1)^ = 0. (27)
d\
If n -=f=- 1 , we can thus conclude that
L(x\x l ) = constant (28)
is an integral of the equations (24).
Now we define the geodesic lines by the variational principle (21) with
(29)
The corresponding Euler equations (24) are
d'l dx*\ 1 dda* dx>
( '
which represent two differential equations of second order for the two
functions # l '(A). Since Lin (29) is homogeneous of the second degree in x l ,
\ 2 , .
-= = constant
is an integral of (30), and by a suitable choice of the parameter A we can
always choose the constant on the right-hand side of (31) to be equal to
1 . This obviously means that we choose the length s of the curve measured
along the geodesic line itself as parameter. (30) and (31) then take the
form
dl _ ki _
ds \ gtk "d) - 2 ~8x> ~fate' gik Us ds { '
It is now seen at once that the curves defined by (30) and (31) also
satisfy the Euler equations (24) with
.
L = ^ *r
which means that the geodesic lines also satisfy the variational equation
i.e. the distance between two points measured along the geodesic line
connecting two arbitrary points has a stationary value.
VIII, 86 THE GENERAL THEORY OF RELATIVITY 231
Geodesic lines, angles between two intersecting geodesic lines as well
as the distance between two points measured along the connecting
geodesic line, are now completely determined by (32), (19), and (18) when
the metric tensor g lk is given as a function of the general coordinates (x l ).
Then we can also deduce geometrical theorems regarding triangles formed
by geodesic lines, etc., on the surface, i.e. the geometry of the two-
dimensional space is completely defined by the quantities g lk (x l ).
87. Determination of the metric tensor by direct measurements.
Geometry in n -dimensional space
The preceding deductions are, of course, completely independent of the
system of coordinates employed If we use another set of curvilinear
coordinates x' 1 connected with the original coordinates x l by a trans-
formation
, ^
we have dx f ' - ^ r/A (35)
c\r k
i.e. the differentials of the coordinates are connected by linear equations.
Therefore if we eliminate the dx< by means of (35) in (IS), ds 2 will also be
a homogeneous quadratic form in dx' 1
ds* = g lk dx l dx k ~- g\ k dx' l dx' k , (36)
where the coefficients g\ k can be regarded as functions of the new co-
ordinates x' 1
Since the geodesic lines are defined by the invariant variational prin-
ciple (21), (29), it is obvious that the differential equations for the
geodesic lines expressed in the new coordinates are obtained from (30) or
(32) by simply replacing g lk and x l by g\ k and x'\ respectively, in other
words, the equations -(30) are covariant or form-invariant. The same
holds for the equation (19).
If it is possible by a transformation of the type (34) to introduce
coordinates X 1 f l (x k ) such that the line element in the new coordinates
assumes the form
ds 2 = (dX l ) 2 +(dX*) 2 = S lk dX l dX k , (37)
the geometry on the surface is called Euclidean. In this case the co-
ordinates X 1 play the same part as do Cartesian coordinates in a
Euclidean plane. The differential equations (32) for the geodesic lines
reduce in these coordinates to the equations
: m
232 THE FOUNDATIONS OF VIII, 87
which have the same form as the equations for straight lines in Cartesian
coordinates. Examples of such surfaces are cylinders and cones which
can be unfolded on a plane without internal deformation. All geometrical
theorems about triangles or other figures on such surfaces are identical
with the theorems of Euclidean geometry, and if we are interested only
in the two-dimensional geometry on the surface, all such surfaces may be
regarded as identical.
In general it is not possible to introduce on the surface such coordi-
nates (Cartesian coordinates) for which the line element assumes the form
(37). In this case the geometry on the surface is a non-Euclidean general
Riemanman geometry. In any case it is possible by means of measure-
ments on the surface to determine the geometry on the surface without
referring to the three-dimensional Euclidean space m which the surface
is embedded. Let us assume that we have introduced an arbitrary
system of coordinates, a' ? , i e. an arbitrary continuous one-to-one corre-
spondence of the set of numbeis (x l ) and the points on the surface. By
means of a measuring-rod we can now measure the distance ds between
the points x l and x l -\-dx l . Since then ds and dx l are known numbers, the
equation (18) represents one equation for the determination of the
unknowns g lk . Since the metric tensor in two dimensions has three
independent components we may thus by performing this procedure
for three suitable line elements starting from the same point x l
completely determine the values of g lk at this point. This can now be
done for any point on the surface, thus obtaining a complete experi-
mental determination of the metric tensor
In this way the geometry on the surface becomes an empirical science
subjected to the limitations arising from the limited measuring accuracy
Now imagine that we heat the surroundings of a gi von point on the surface
so that the measuring-rods are dilated when inserted at this point. If
we neglect this dilatation we shall, by means of the method described
above, find wrong values for the components of the metric tensors.
Since, however, the thermal dilatation is different for measuring-rods
prepared from various materials, it is easy to correct for this error and to
find the 'real' values for the components of the metric tensor. On the
other hand, it is obvious that, if all measuring-rods in the neighbourhood
of a given point for one or another reason were dilated at the same rate
independently of the material of which they are made, it would be
impossible to observe this dilatation and in an unambiguous way to
correct for it. Therefore there is no well-defined meaning in the state-
ment that such an expansion of the standard rods has taken place, and
VIII, & 87 THE GENERAL THEORY OF RELATIVITY 233
from a physicist's point of view the metric tensor and the geometry
obtained by measurements with the natural standard rods must be the
'real' geometry on the surface.
All the considerations of this paragraph for the two-dimensional case
can now be immediately generalized to spaces of 3, 4, or n dimensions.
The only difference is that the points in an n-dimensional space are
characterized by n coordinates x l and that all indices in the preceding
equations now can take on the values from 1 to n. The length of a line
element and the angle between two line elements are then again given
by (18) and (19), respectively, and the'geodesic lines are defined by the
n equations (30) or by means of the variational principle expressed By the
equations (21), (22), (29).
88. General accelerated systems of reference. The most general
admissible space -time transformations
In 84 we have seen that the spatial geometry in a uniformly rotating
system of reference is non -Euclidean and also the temporal description
is more complicated than in the systems of inertia. This may be regarded
as an effect of the gravitational field present in the rotating system of
reference According to the principle of equivalence we must therefore
expect that a gravitational field m general will manifest itself not only
by the presence of gravitational forces (centrifugal forces, Coriolis forces,
gravitational attraction between masses, etc ), but also in the results
of space and time measurements.
Let us start again from a system of inertia / with the usual space and
time coordinates (X, Y, Z, T) To each set of values of these variables
corresponds a certain event which is represented by a point in (3+1)-
space with coordinates
X 1 = (X,Y,Z,cT). (39)
These coordinates differ from the coordinates defined in (IV. 2) only in
that we have dropped the i in the fourth coordinate. The four-dimen-
sional line element (IV. 26) thus takes the form
efo a = dX 2 +dY*+dZ 2 -c* dT* - G lk dX>dX k , (40)
f for i ^ k
where O tk = j 1 for i == k =- 1,2, 3 (41)
[ 1 for i = k = 4.
Instead of the pseudo- Cartesian coordinates X l we shall now introduce
general 'curvilinear' coordinates x l in four-space by means of the trans-
formations / t ,VL-\ /A*\
x l x l (X k ), (42)
234 THE FOUiNDATlOJNS Ob Vlll, & 88
where the x l ( X k ) are arbitrary continuous and differentiate functions of
the variables (X k ). The transformations ( 1 ), (3), ( 10) obviously represent
a special case of (42). By differentiation of (42) we obtain
3~.i
U*' i-iri* A i n^rl* (d.^
From the relations reciprocal to (42) we obtain in the same way
dX* = ~~ dx k - Afc dx k . (44)
Using (44) in the right-hand side of (43) gives
dx l A}& l k dx k .
Since this equation is to hold for arbitrary dx l , we must have
*"^ (45,
for i = k.
In the same way, substituting from (43) in (44), we obtain the
equations A} A' A = Sj^ (45')
Elimination of dX l m (40) thus gives the following expression for the
interval: d8* = g lk dx*<h* 9 (46)
ffi* =- flu = 0, m ^ A r ^ A ; A k~ ^ A (47)
(Greek indices run from 1 to 3, Latin indices from 1 to 4).
The system of coordinates (JL I ) determined by the transformations
(42) also determines a definite system of reference. Defining a 'point
of reference' as a point with constant values for the three space -
coordinates (x l ), the system of reference corresponding to the system of
coordinates (x 1 ) can be defined as the collection of all points of reference.
In general such a system of reference will, of course, not be rigid, since the
different reference points may have largely varying velocities relative
to 7. The motion of the different points in the system of reference will
therefore in general be analogous to the motion of a fluid and we shall
confine ourselves to such transformations (42) for which the correspond-
ing system of reference can be pictured by a real fluid. This means that
the velocities of the points of reference relative to the system of inertia /
must always be smaller than c. Since for any point of reference dx l =
we obtain from (44) for the velocity components v l of such a point relative
to/> V dX' Aj . .
(48)
VIII, 88 THE GENERAL THEORY OF RELATIVITY 235
If this velocity is to be smaller than c, we must have
\ = V ~T < l or A 4 A 4 Aj Aj < 0. (49)
c 2 c 2
In order that the system of reference defined by (42) may be physically
realizable, the admissible space-time transformations must satisfy the
condition (49) which, on account of (47), involves
044 < 0- (50)
At every point of reference we imagine a coordinate clock to be inserted
showing the time t x*/c, and we must now further demand that the
time description obtained in this way shall give a reasonable causal
description of physical phenomena. Thus a real signal which is emitted
from a point of reference (x l ) at the time t must arrive at a point of
reference (x^dx 1 ) at a time t+dt with positive dt. Since signals can at
most have the velocity c relative to /, the line element
must be smaller than or equal to zero for two adjacent points on the
time track of the signal. In the system of coordinates x l this means
ds 2 = g lk dx l dx k ^ 0.
In other words, any two events which are simultaneous in the system
of coordinates (x l ), i.e. for which dx* = 0, cannot be connected by a
signal, so that in this case we must have
g lk dx*dx k > 0,
or, since dx* = 0, g lK dx l dx K > 0. (51)
This inequality must now hold for arbitrary dx l , which shows that the
quadratic form g lK dx^dx" must be positive definite. The necessary and
sufficient condition for this to be true is that all subdeterminants of
the scheme of numbers g lK are positive.
The admissible transformations (42) must therefore be such that the
corresponding g lk satisfy the conditions
0u 012 013
<7 U >0,
9 " 9iK
t/Ki VKK
031 032 033
0, <? 44 < 0, (52)
where t and K may be any of the numbers 1, 2, 3 (no summation over t
in U ).
236 THE FOUNDATIONS OF
From (52) it follows that the determinant
011 012 013 014
= I0,*l =
022 023 024
< 0.
VIII, 88
(53)
031 032 033 034
041 042 043 044
If we introduce another system of coordinates x' 1 by means of the
transformation equations
x' 1 = x' l (x k )
t' 1 ^ aju dx k -= r dx k
dx k
dx l = &[ dx' k =
we have, by analogy with (45), (45'),
fa*
dx' k
(54)
a
Expressed in the new coordinates the interval will again be a homo-
geneous quadratic form
ds 2 =- g lk dx l dx k = g\ k dx' f dx' k , (56)
where g lk = g' kl - g lm &[ cx J (57)
are new functions of the coordinates x' 1 . The relations reciprocal to
(57)are ^ = 0; n xr, (58)
which can easily be verified by substituting from (57) in the right-
hand side of (58) and using (55). The transformations (54) must be such
that the new functions g' lk again satisfy inequalities of the form (52).
Tn general the system of reference R' defined by the system of co-
ordinates (x/ 1 ) will be different from the system of reference R corre-
sponding to the coordinates (x 1 ), but if the transformations (54) are of
the form x = x(x] }
, (59)
x'* = x'*(x*) = /(a*) j V
where the space coordinates x' 1 are functions of the spatial coordinates
x* only, the systems of reference R' and R are identical. For in this case
the transformation simply implies another notation for the points of
reference in R together with an arbitrary continuous change in the rate
and setting of the coordinate clocks. While each system of coordinates
(x 1 ) corresponds to one and only one system of reference R, we can
always in a given system of reference introduce an infinite number of
VIII, 88 THE GENERAL THEORY OF RELATIVITY 237
different space-time coordinate systems which are connected by trans-
formations of the form (59). The coefficients a*. and a k corresponding
to the special transformations (59) obviously satisfy the conditions
(59')
In general the gravitational fields in different systems of coordinates
will be different, but for physical reasons it is convenient to consider
the gravitational fields in all systems of coordinates connected by (59) as
identical, since all these systems of coordinates correspond to the same
system of reference. In different systems of reference, however, there
will in general be different gravitational fields. Jn this respect the various
systems of inertia are exceptional, since they all have a vanishing
gravitational field.
89. Space and time measurements in an arbitrary system of
reference. Experimental determination of the functions g ik
Let us now consider an arbitrary system of reference R into which we
have introduced a certain system of coordinates (x l ). Consider in par-
ticular two points of reference A and B in this system with the space
coordinates (x l ) and (x L -}~dx L ), respectively The spatial distance da
between A and B at the time / ---- JT*/C can now be measured by means of a
standard measuring-rod connecting the points A and B and at rest
relative to A In the limit of very small dx L , B will also be practically
at it\st relative to the measuring-rod In order to express da in terms of
the functions g lk we introduce the system of inertia 7 relative to which
the point of reference A (arid approximately also B) is at rest at the
time t If X L denote the pseudo-Cartesian space-time coordinates in 7,
the transformation from 7 to R is given by the equations (42)-(48).
However, since 7 is the rest system of the point A at the time considered,
we have, on account of (48),
A^ - (60)
at the point A and the time t. The differences dX L between simultaneous
values of the Cartesian coordinates of the points A and B at the time of
the measurement are obtained from (44) by putting
i ~- i and dx* = dt = 0.
Hence dX L = A L K dx. (61)
On account of (60) the term A 1 4 dx l in (44) would vanish anyhow, even if
dx* / 0, which means that the values of dX L would be approximately
the same as for positions of A and B which are simultaneous relative to 7.
238 THE FOUNDATIONS OF VIII, 89
According to the general assumption formulated on p. 223, the
standard measuring-rod in the system R has the same length as the
measuring-rod in /; therefore, da 2 is simply
Using the expression (61) for dX l we see that da 2 is a quadratic form
in the differentials (dx L ), i.e.
da 2 = y lK dx'dx" (62)
with y lK = A* A*.
Now we get from (47) and (60)
Hence A 4 ^ g ' 4
A,_ -
where we have put y t EE ~~J^ . (63)
Furthermore, we obtain from (47) for i t, & :
\ A K g lK ~\-A l A K ,
which leads to the following expressions for the spatial metric tensor
YlK = QlK + YlYK: ( 64 )
The metric tensor y lK which determines the spatial geometry in the
reference system R will thus in general not simply be equal to the spatial
part g LK of the four-dimensional metric tensor g lk . This is the case only if
0*=0 or y l = 0. (65)
In a system of coordinates where the equations (65) are satisfied at
every point in 4-space, the time axis is everywhere orthogonal to the
spatial coordinate curves. Such a system will therefore be called time-
orthogonal.
Now consider a standard clock C inserted at rest at a given reference
point A of the system of reference R. The line element of the time track
of this clock is then given by
ds* - 44 (dx*) 2 (66)
since dx l = for a clock at rest. While t ~ x*/c denotes the time shown
by the coordinate clock at A, the increase dr in the time of the clock C
is given by ^ = _ c ,
VIII, 89 THE GENERAL THEORY OF RELATIVITY 239
(cf. 37). This follows at once from the in variance of ds 2 if we introduce
the system of inertia 7 which is momentarily at rest relative to the clock
C. Expressed in terms of the space-time coordinates (X Qi ) of this system,
ds* becomes ^ _ _^ (d?70)2
since dX Ql = 0. Further, since the time T Q itself is given by standard
clocks at rest in 7, we have dT Q dr , and (66) can be written
drl = -<7 44 dt*. (66')
Consequently the function ^/(~g^) determines the ratio between the
rates of the standard clock C and the coordinate clock at the place con-
sidered. The quantity </ 44 may thus be obtained experimentally by
measuring the ratio of the rates of the two clocks in question. If we do
this at each reference point and at all times, we get g u (x l ) as a function
of the space-time coordinates (x l ).
The metric tensor y iK which, by (62), determines the space geometry
in S can now also be obtained by direct measurements, applying a
method similar to that explained for the two-dimensional pase in 87.
To determine the six independent components y lK we simply need to
measure the lengths of six properly chosen hne elements (dx L ) at each
point and at every time.
Now, if we could also find a procedure which would allow us to measure
the quantity y t , the metric tensor g lk would be completely determined . To
this purpose we consider a light signal which starts at the reference point
A with space coordinates (x l ) at the time t and arrives at a neighbouring
point B: (x l -\-dx l ) at a time t-{-dt, say. The time track of this signal is
characterized by the equation
ds* = g lk dx l dx k = 0, (67)
for in the system of inertia / the velocity of the signal is c, which
means that the invariant
ds 2 = O lk dXidX* = dX*+dY*+dZ*-c* dT* = 0.
The equation (67) may also be written
g iK dx'dx+2g L t dx^dx^+g^ (dx*)* = 0,
or, by means of (63), (64), and (62),
da*-(<y L dx^(y K dx-)+^(-g^} 7l dx^dx^+g^ (dx*)* = 0.
If we divide this equation by dt 2 we obtain
w* = { yt ^_cV(-<744)} 2 , (68)
240 THE FOUNDATIONS OF VIII, 89
dx L
where the w l = - = wn l (69)
at
are the components of the velocity of the light signal m the direction
n l and
w = Tt = V(y^'^) (69')
is the magnitude of this velocity. In the first place, we see from (68) that
the light velocity w depends on the direction of propagation n l of the
signal if y t ^ in the system of coordinates considered.
In fact, using (69) in (68) we get
where y lK nlnK ~ *> 1 - e - (yt^ 1 ) 2 < l -
On the other hand, if we solve the equation (70) with respect to (y L n l ),
a measurement of the velocity of light in three different directions n l
will allow us to determine the three quantities y t . In this way it is
possible in principle to determine all the quantities y llc , y t , </ 44 , i.e.
the complete metric tensor g lk > by experiments.
90. The spatial geometry in the rotating system of reference
Considering again the rotating system of coordinates introduced in
84, we obtain from (1), (2), (3), and (10) the transformation equations
(42), or rather the corresponding reciprocal equations, in the form
X = rcos(#+o>Z), Y = rsm(&+cot)
Z^z, T = t . (71)
By differentiation and introduction of (dX,dY,dZ y dT) into the ex-
pression for the interval
we get
ds 2 = dr*+r 2 ^ 2 +d2 2 +2cur 2 d&dt-(c*-rW) dt* = g lk dx l dx k . (72)
Thus we have
VIII, 90 THE GENERAL THEORY OF RELATIVITY 241
all other components of g lk being zero. Hence, from (63) and (64),
y t = |0, - T 2 2 , OJ = ^ 2 ~ 8 l2
= 1 - r 2 = x . (74)
y lK = for L -^ K
Thus if we calculate the spatial line-element da 2 by means of (62) and
(74) we come back to the expression (7). Further, (66') together with the
expression (73) for j/ 44 leads back to the equation (9).
The space geometry defined by the line-element da 2 = y lK dx l dx K is
non-Euclidean. In the plane z = Z = 0, i.e. on the rotating disk, we
have only the two coordinates (x l ,x 2 ) ~ (r,$) and the geodesies are
determined by the equations (32)
r
da l 2 dx l da da
dx l dx K
They define the curves of shortest distance as measured by measuring-
rods at rest on the rotating disk. In our case we get from the second
equation (75 a) and (74)
d
and by integration r ST^# = a > (76 a)
1 r z w 2 /c z
where a is a constant, and & d&jdv. Hence,
fl-rW/c*\ (7e . ,,
V = Oil - g -- 1' ' '
Further, from (756)
r 2 / 1 l " w _ * \
TT dr r V \ c 2 r 2 /77 ,
Hence, . TO = -* = i-^^-S T-?V -> ( 7T )
which by integration gives r as a function of & for the geodesies.
3595.60
242 THE FOUNDATIONS OF VIII, 90
If the constant of integration a is zero, we get from (76 a') & = 0, i.e. the
radius vectors with # = constant are geodesies. In Fig. 17 we have
given a picture in a Euclidean plane of some of the geodesies on the
rotating disk. In this picture the points on the disk with the coordinates
(r, &) are depicted as points with the polar coordinates (r, #). The curves
D
FIG 17
in this picture are therefore simply the curves in the fixed JfF-plane
with which the points on the geodesies of the rotating disk coincide at
the time t = T 0. The radius r* of the disk is defined by (4), i.e.
r* = c/co. (78)
For the geodesies OB and OA we have a = 0. The geodesic AB which
starts at the point (/* , 0) at right angles to OA, i.e. with r = 0, corre-
sponds to an a ^ which follows from (76 a'), (766):
r 2 . r
_ / __ d _ __
( 1 -r a>'c) ~ ( 1
With increasing values of & the radius vector r increases also with a rate
determined by (77), and it is seen at once that dr/d& is larger than in the
limit c -> oo, where we should have
dr r(r 2 a 2 )*
- - = - . (80)
air a
The broken straight line in Fig, 17 is the curve determined by (80) which
would be the geodesic if the geometry on the rotating disk were Eucli-
dean. When r approaches the value r* = c/a> we get # -> from (76 a'),
which means that the geodesic A B has OB as tangent at the point B.
Consider two geodesies x\ = a:i(or 1 ), xfa = ^(o^) which go through
VIII, 90 THE GENERAL THEORY OF RELATIVITY 243
the same point. The angle between the two curves at the point of
intersection is determined by (19). Thus, on account of (74) and (76),
a >W di
cos 6 = ^-=
1 T Ct) 1C
n ~'" Z -=*.} + '-
r 2
(81)
where c^ and 2 are the values of the constants a for the two geodesies
in question. This expression can be used everywhere except at the
point 0.
Now consider the triangle OAB. At O the angle between the two
sides of the triangle is equal to 6 O = & B < \TT, for in the centre of the
disk the geometry is Euclidean. The angle between the geodesies AO and
A B at A is found from (81) by putting
thus,
i.e. A = \7T.
Finally, we get the angle 6 B between the geodesies BA and BO from
(81) by putting x = 0, a 2 = -a, r = r* = c/a>. Thus we get
Hence the sum of the angles in the triangle OAB is
i.e. the sum of the angles in a triangle on the rotating disk is smaller than
TT. Only if the triangle is close to the centre is the sum of the angles
approximately equal to the Euclidean value TT. There are even triangles
on the rotating disk for which the sum of the angles is zero. If, for in-
stance, #B = |TT, we see that the triangle CBD formed by the geodesies
CB, BD, and DC has a vanishing sum, i.e.
QCBD = -
Thus the sum of the angles in a triangle on the rotating disk can have all
values between and ir. The spatial geometry as determined by ob-
servers on the rotating disk is the same as on a surface of negative
curvature in a three-dimensional Euclidean space.
244 THE FOUNDATIONS OF VIII, 90
Sometimes it is convenient to use a different set of space coordinates
(x,y,z) in the rotating system connected with the cylindrical co-
ordinates (r,#, z) by the equations
x rcos#, y = rsm&, r 2 = x 2 -{-y 2 . (82)
In these coordinates the line element (72) takes the form
ds 2 = dx 2 +dy 2 +dz 2 +2aj(-ydx + xdy)dt- /l-^c 2 ^ 2 . (83)
91. The time tracks of free particles and light rays
Consider a material particle which is moving freely under the influence
solely of the gravitational fields in an accelerated system with the
coordinates (x l ). Since these fields are supposed to be non-permanent,
they can be transformed away simply by introducing the pseudo-
Cartesian coordinates (X l ) of the system of inertia / from which we
started in 88. In this system the motion of a free particle is uniform,
i.e. its time track is a straight line defined by the equation
-
where A is an arbitrary parameter. In other words, the time track of a
freely falling particle is a geodesic in 4-space. The geodesies are de-
fined by the variational principle (21), (29), (33), where the indices &, k
now are running from 1 to 4. If we introduce the proper time r = s/ic
as parameter, the vanational principle (33) states that the variation
f
J
-- (85)
dr \ l dr 2 dx* dr dr ( >
must be zero for all variations 8x l which vanish for r = r 1 and T = r 2 .
This leads to the Euler equations (32) or
d( dx k \_ldg kl dx*dtf dtfdx* _
dr\ 9lk ~&r) - 2^-fo dr* 9lk ^^ ~~ ' ( b)
In a pseudo- Cartesian system of coordinates the metric tensor is
G lk as defined by (41). The equations (86) then reduce to (84), and (85)
reduces to the equation (V. 102) used in 59.
Now consider a light ray in empty space. In the system of inertia /
its time track is again given by (84), but with the extra condition that
ds 2 = G lk dX l dX k = 0. The time track of a light ray is thus a geodesic of
VIII, 91 THE GENERAL THEORY OF RELATIVITY 245
zero length, and we can therefore not use the length as a parameter.
Instead of (86) we thus have in an arbitrary system of coordinates (x l ) the
equations (30) and (31) with the constant in (31) equal to zero, i.e.
dt dx k \ __ 1 dg kl dx k dx l dx l dx* __
dtif* d\) ~ 2W rfA dA ' 9lk ~d\~dX ~ ' ( n
where x l == a; 1 (A)
may be any parametric representation of the time track.
92. The dynamical gravitational potentials
In 89 and 90 we have seen that the gravitational field in an arbitrary
accelerated system of coordinates (x l ) influences the space and time
measurements as made by standard measuring-rods and standard clocks.
The space geometry, for example, is determined by the spatial tensor y tK
defined by (64). We shall now determine the quantities which describe the
dynamical action of the gravitational fields. For this purpose we shall
use a test body of arbitrary mass, which is placed at rest at the point
of our system of reference at which we wish to measure the gravitational
field. The acceleration imparted to this particle by the gravitational
field then determines the strength of the field.
From (86) we get, for a particle which is momentarily at rest, taking
' = * = 1. 2, 3. d *x* d I dx*\ 1 dgjdx'
(88)
Further, we get by means of (64), (63), and (62) for a particle in arbitrary
motion
c 2 dr 2 = ds 2 = g lH dx L dx K +2g^ dx l dx*+g^ (dx*) 2
where r is the proper time, and the
are the components of the velocity of the particle.
Dividing (89) by (dx*) 2 and solving this equation with respect to dx*/dr
gives dx* i / v u' \ 2 }-*
~- - c (-M 1 - :^Vl) - U2/C2 \ ' (90)
dr [ \ <v( 044); J
246 THE FOUNDATIONS OF VIII, 92
da
where u ~Ji
at
is the magnitude of the particle velocity.
Hence g^ ~ = cy t | 1 1 ,~ r I -f- ^ 2 /c 2 {744 J >
and for a particle which is momentarily at rest we get after some cal-
d
and
Using (91) and (90) in (88) we thus obtain
i inn\
- (92)
The left-hand side represents the gravitational acceleration imparted to
the test particle, ~
being the covariant components of the acceleration expressed in the
curvilinear coordinates of our system of reference (see 100). Thus the
dynamical action of the gravitational field is described by the functions
044 and y t .
If we put 44 _ /i + \ (94)
we get from (92) a L = ~^- c I ! + - }^- i . (95)
dx l AJ \ c 2 / dt
On account of the analogy of this expression with the expression for
the electric force on a charged particle at rest in terms of the electro-
magnetic potentials, the quantities x an d y L will be called the gravita-
tional scalar and vector potential, respectively. The scalar potential
X has been normalized so as to give <jr 44 the value 1 of the special theory
of relativity for a vanishing potential.
If y l is time-independent, the gravitational acceleration is simply
equal to the gradient of the scalar potential :
a t =- 3 . (96)
VIII, 92 THE GENERAL THEORY OF RELATIVITY 247
This is, for instance, the case in the rotating system of coordinates con-
sidered in 90, and from (73) and (94) we get in this case
x = -r 2 co 2 . (97)
Thus the gravitational acceleration of a particle of zero velocity lies in
the direction of increasing r and is equal to ro> 2 . This is in accordance
with the usual expression for the centrifugal force.
93. The rate of a moving standard clock in a gravitational field
From (90) we get for the proper time of a particle moving in the gravi-
tational field described by the potentials (y t , x)
-
Since r is the time measured by a standard clock following the particle
in its motion, (98) gives the rate of the moving standard clock compared
with the rate dt of the coordinate clocks of the system considered. If
the system of coordinates is time-orthogonal, we have y t = and
(99)
The formulae (98) and (99) are the generalizations of the formula (II. 38)
and express the retardation (or advancement) of moving clocks in the
case where gravitational fields are present.
For a clock at rest in our system of reference we have
) (100)
in accordance with (66') and (94). The rate of a standard clock thus
depends on the scalar gravitational potential at the place where the
clock is situated; it is lower at places of small gravitational potential,
On the rotating disk we have, according to (100) and (97),
dr = d^/(l + 2x/c a ) - dtj(l-r*a>*/c*), (101)
i.e. a standard clock far from the Centre has a slower rate than a standard
clock placed at the centre which simply shows the time t = T. This
retardation of a standard clock at a place with r > will be differently
interpreted by observers in the fixed system 7 and in the rotating system
S. An observer in / will explain the retardation by the motion of the
particle. In this system we have no gravitational field, i.e. x but
the velocity of the clock is u = ra>. An application of (99) in / thus leads
to the formula
dr Q = dt<J( 1 u*/c*) -= dt<J( 1 rW/c 2 ). ( 1 02)
248 THE FOUNDATIONS OF VIII, 93
On the other hand, in S the velocity u 0, but we have a gravitational
field x = ^r 2 o> 2 , and (99) again leads to the same expression (101) or
(102). An observer in 8 will thus explain the retardation by the action
of the gravitational field present in the rotating system.
94. Transformation of coordinates inside a fixed system of
reference
Let (x l ) be a set of space-time coordinates corresponding to a certain
system of reference jR. By the transformation (59) we may then intro-
duce new space-time coordinates inside the same system of reference R.
The transformation (59) simply introduces a new numbering of the refer-
ence points in R together with an arbitrary change in rate and setting
of the coordinate clocks This, of course, cannot give rise to any change
in the spatial geometry in R determined by measurements with standard
measuring-rods, i.e. da as defined by (62), (64), and (63) must be invariant
by the transformation (59). A formal proof of this statement is given in
Appendix 4.
Since the system of reference is unchanged by the transformation
(59), the gravitational field must also be regarded as unchanged. The
gravitational potentials (x,y t ) defined by (94) and (63) will, however,
be transformed in accordance with (57). In this respect a transformation
of that kind is thus analogous to the gauge transformation (V. 23) of the
electromagnetic potentials by which these potentials are changed without
any influence on the electromagnetic field derived from the potentials.
In many cases it is possible by such a 'gauge transformation' of the
gravitational potentials to give the potentials a particularly simple form.
In the first place, it is always possible by a transformation of the type
x'' = a 1 ; z' 4 - x'*(x*) = f(x*) (103)
to ensure that the scalar potential in the system (x' 1 ) vanishes. We need
only choose the new time variable t' = x'*/c so that the new coordinate
clocks have the same rate as standard clocks at rest. On account of (66')
and (100), this is obtained by putting
t
f = T = J V(-
(104)
where the integration is performed for constant values of the space
coordinates (X L ), and ^(x l ) may be any function of the space coordinates.
With this choice of the new time variable we get, since (100) must hold
VIII, 94 THE GENERAL THEORY OF RELATIVITY 249
also in the new system,
dt' =-- dr - dt' III + -*-) = dt'J(-<f u ),
V \ c /
i.e. x' = 0, fik=-l. (105)
A more useful simplification would, however, be obtained if the vector
potential could be 'transformed away' by a transformation of the type
considered, for this would mean that the new system of coordinates
is time-orthogonal and all formulae are considerably reduced Let us
therefore try to find a transformation (103) such that
y\ - 0. (106)
Since the interval can be written in the form (89) in every system of
coordinates, and since da 2 is invariant under the transformations (103),
we must have
^.'4 = !/p^ 4 _ r t - (107)
must be the total differential of the function /(x' ) in (103). Thus we see
that the space-time derivatives of the function /must have the following
ratios .
j-f J^ J J ?i_ - -2^? __ . _ 73 __ __! QAON
'
This is equivalent to the validity of the three equations
= for i=l,2,3. * (109)
V ;
These conditions could also be obtained from (106) by mep,ns of the trans-
formation properties of the gravitational potentials (see Appendix 4).
Now the simultaneous differential equations (109) have a solution
only when the following compatibility conditions are satisfied. If we
multiply (109) by the operator
and subtract the equation obtained in this way from the equation with
250 THE FOUNDATIONS OF VIII, 94
and K interchanged, we get after a simple calculation the condition that
the quantity
1-
must be zero, i.e. o> tAC = (HI)
for all values of t and K. (Ill) together with (110) represent the general
condition which the dynamical gravitational potentials must satisfy in
order that the vector potential may be made to vanish by a transforma-
tion of the type (103).f
In the case of the rotating system of coordinates considered in 90,
we see at once from (110), (73), and (74) that the only non-vanishing
components of o> tK . are
aj l2 = o> 21 = r^_-_, (112)
which are different from zero for r > 0. The condition ( 1 1 1 ) is therefore
not fulfilled in the rotating system and it is not possible by a simple
change of rate of the coordinate clocks to introduce a time-orthogonal
system of coordinates in this system of reference.
The gravitational field in a given system of reference R is called
stationary if it is possible, by a suitable choice of the space-time co-
ordinates in R, to ensure that all components of g lk , i.e. y LKJ #, and y t ,
are independent of the time variable. If simultaneously we can obtain
y t = 0, the gravitational field is called static. The gravitational field
in the uniformly rotating system of reference is thus stationary.
95. Further simple examples of accelerated systems of reference
Let X 1 = (X,Y, Z,cT) again be pseudo- Cartesian space-time co-
ordinates in a system of inertia /. The Galilean transformation (I. 2)
y X wT 7 it y 2 7i f T n 1 *U
b -*\. V J. ) if JL , A> J , I JL \LlOf
then defines a new system of reference which obviously is the system of
inertia /' moving in the direction of the X -axis with the velocity v relative
to /; for each reference point (x>y, z) = constant is moving with the
f This condition was found by Weyssenhoff, see J. v. Weyssenhoff, Bull. Acad.
Polonaise. Sor. A, p. 252 (1937) ; soo also Z8. J. Phys. 95, 391 (1935) ; ibid. 107, 64 (1937).
VIII, 95 THE GENERAL THEORY OF RELATIVITY 251
same velocity v. If we put (x l ) = (x,y,z,ct) the interval takes the form
ds* = g lk dx*dx k = O ik d
i.e. U = 02 2 = g 83 = 1, U == 41 = v/c, 44 - -(l-i> 2 /c 2 ),
all other components vanishing. The system of coordinates (a;*) is there-
fore not time-orthogonal and from (63), (64), and (94) we get
V33
for t
Since the gravitational potentials are constant the gravitational field
determined by (95) is, of course, zero in /', and the potentials may be
transformed away by a transformation (59) with / of the form (104).
Introducing new coordinates X' 1 = (X 1 \ Y', Z',cT f ) by
X' x Y' 11 Z' z
'- *- (116)
r - A-
we et
The coordinates (X /x ) are pseudo- Cartesian coordinates in 7 ; , they are
connected with the (X 1 ) by the special Lorentz transformation.
As a further example we consider the accelerated system
(x l ) = (x, y, z, ct)
defined by the transformation
X = x+\gt*, Y = y, Z = z, T = t. (118)
Each reference point in the system (x 1 ) has a constant acceleration g in
the direction of the JT-axis relative to /. A simple calculation gives
ds* = dx 2 +dy*+dz*+2gt dxdt-c* dt*(l -^1 (119)
\ c 2 /
i.e. ffu = fe = 033 = !> (7l4 = 041 = ^/C, 044 = f 1 ""
all other components g tk being zero.
252 THE FOUNDATIONS OF VIII, 95
Hence,
Yi = I fl*_ -.0,0), x = - V
. (120)
i ^ r / I
= 733 ~
This system of coordinates corresponds to a physically realizable system
of reference only for t T < c/g, for only in this case will the velocity
gt of the reference points relative to / be smaller than c and gr 44 < 0.
The space geometry in this system is defined by the spatial line element
Jr 2
da 2 - y lK dx<<dx = -J- + dif+dz 2 . (121)
1 g*t*/c*
It is non-Euclidean on account of the Lorentz contraction of the measur-
ing-rods in the moving system Since the potentials do not depend on the
space coordinates the quantities w lK in (110) are zero, which means that
the vector potentials can be made equal to zero by a transformation
(103). Integrating the equations (109) we find that the transformation
leads to the desired result, i.e. if ds 2 is written in the form (89) we get m
the new coordinates
e 2gXIc*
</44 =- ~ 12^2 = ~ in^yaZTa
J ' y ' . (123)
^9/ ' I 1 \ I
/ A ' '
It is easily verified by direct Calculation that (122') is identical with
(119) when use is made of the transformation equations (122).
The gravitational field in the accelerated system considered is non-
static. According to (121) even the space geometry is time-dependent,
i.e. the distance between two neighbouring reference points depends
on the variable t. This is also evident, since the measuring-rods in the
accelerated system on account of the increasing velocity gt relative to /
will undergo an increasing Lorentz contraction. Although our system of
reference is rigid from the point of view of an observer in /, an observer in
the accelerated system itself will find that the system of reference points
is dilated in the direction of the #-axis,
VIII, 95 THE GENERAL THEORY OF RELATIVITY 253
For small values of t and x, i.e. if we retain only terms of the first order
in gt/c and gx/c z , we get t f - t(l-gx/c*), g^ = -(l+2gx'/c*) y x ' = 9*',
and from (96) o /
a '=-0i=<-0>')' (124)
i.e. the gravitational field is constant in this region.
96. Rigid systems of reference with an arbitrary motion of the
origin
A system of reference is called rigid if the distance between two refer-
ence points, as measured by standard measuring-rods at rest in the
system, is constant in time. Thus the uniformly rotating system dis-
cussed in 90 is rigid, while the system considered at the end of 95 is not.
Consider now a particle in arbitrary motion relative to the system / with
the coordinates X^ = (X,Y>Z,icT). Its time track may be described
by the equations
^ =/,(*), (1^5)
T being the proper time of the particle. We shall now try to introduce
a system of coordinates (x l ) = (x, y, 2, rt) which is the relativistic analogue
of a classical rigid frame of Cartesian axes following the particle in its
motion, so that the particle is constantly situated at the origin of this
frame of reference, and the space axes have constant directions. In
46 we have determined the successive systems of inertia $'(r) which are
momentary rest systems of the particle and which are successively
obtained by infinitesimal Lorentz transformations without rotation of
the spatial axes. The transformation from the fixed system (XJ to the
coordinates x[ in S'(r) is then given by (IV. 140), where the coefficients
a lfc (r) are determined by the differential equations (IV. 136, 137).
Now defining the system (x l ) by putting
x* =- x' t , 4 0, t = r (126)
in (IV. 140), we find that simultaneous positions of the reference points
in the system (x l ) at the time / coincide with the simultaneous positions
of the reference points in S'(r) at the time x^ = 0, and the coinciding
reference points have the same values for the spatial coordinates in these
two systems of coordinates. The transformation connecting the variables
(X % ) and (x l ) are thus
X % =/&)+**(*)> ( 127 )
where the coefficients a lk are the solutions of (IV. 136). These equations
are completely determined by the given motion (125) of the particle
254 THE FOUNDATIONS OF VIII, 96
which lies permanently at the origin x 1 = of the system (x 1 ). Differen-
tiation of (127) gives, by means of (IV. 136),
dXi = [AW+a^W] dt+dx* Kl (t)\ (128)
If we use (128) in the expression for the interval we get, on account
of (IV. 138),
ds 2 = dX l dX l = dx*-\-dy
x * = (x^ct) = (x,y y z,ct)
where g K = g K (t) = * Kl U t = * Kl f t ( 1 30)
are functions of t only, which are completely determined by the motion
of the origin of our system of coordinates (x 1 ) relative to the system (XJ.
The quantities g K are equal to the components of the acceleration of the
particle in its momentary rest system S'(t) (cf. (IV. 42, 42')).
The system of coordinates (x 1 ) defined by (127) is time-orthogonal.
The corresponding system of reference is rigid, for the distance between
two reference points (x,y,z) and (x-\-dx y y-\-dy,z-{-dz) is given by
da 2 = dx*+dy*+dz 2 . (131)
Thus the space geometry is even Euclidean and (x, y, z) are Cartesian
space coordinates. The vector potential is zero and for the scalar
potential we get
(132)
Its dependence on the space variables is thus always of the same type,
only the coefficients g = g(<) will be different for different motions of
the origin. The gravitational field is determined by (96) and (132):
a = --gradx - _g(i + (g. x )/c 2 ). (133)
Hence, in a region around the origin, where
(a.xKc,
the gravitational field is homogeneous, the gravitational acceleration
a g g(<) being a function of t which is uniquely determined by
the motion of the origin x ?= relative to /.
The rate of a standard clock at rest at the point x as given by (100) and
(132)IS dr 0== ^(l + (g.x)/C 2 ). (134)
At the origin x = we have T O = t, i.e. the coordinate clock at this place
is a standard clock.
VIII, 96 THE GENERAL THEORY OF RELATIVITY 255
It may be shown that the type of accelerated systems considered in
this section together with the rotating system of 90 are essentially the
only possible rigid systems of reference in the case of non-permanent
gravitational fields.
97, Rigid frames of reference moving in the direction of the
X-axis
When the origin O of the system (x l ) is moving in the direction of the
Jf-axis of the system (XJ, the coefficients a lk are given by (IV. 154, 147),
and the transformation equations (127) reduce to the equations
i
X = c j smh 6 dt+x cosh 0(t)
(135)
f smhB(t)
J c
o
which are also obtained from (IV. 155) by the substitution (126). For
the vector (130) we get, by means of (IV. 154, 146),
g- (0,0,0), g(t)^c d /-> (136)
at
Hence, from (129),
(137)
This is also easily obtained directly by differentiation of (135) and
substitution in the expression for ds 2 in terms of the differentials
(dX^, In this case the gravitational field is parallel to the x-axis. The
conditions (52) are satisfied everywhere except on the plane x = c 2 /<7,
where gr 44 becomes equal to zero.
In particular, if the motion of the origin is a hyperbolic motion with
the constant rest acceleration g, we have, according to (IV. 159),
i.e. g(t) = c - = g = constant
at /
In this case the gravitational field is static, the gravitational potential
being 2
256 THE FOUNDATIONS OF VIII, 97
The transformation equations then reduce to
X = - (cosh g - - l\ + x cosh g -
g\ e ) c
Y = y,
(140)
9
as seen from (135) and (138) or from (IV. 160) and (126)
By elimination of the variable t in (140) we get
X = ? {[(l+^)+^W-J}|
Y~y, Z = z )
Thus we see that, relative to /, each reference point with constant values
for x, ?/, z performs a hyperbolic motion in the direction of the ^-axis
starting at the point X -- x, Y ?/, Z = z at T = 0, with zero velocity
The acceleration in tlus hyperbolic motion is
Y = f/ (142)
1 -f yx/c 2
(see 29), and the velocity at the time T is
dX gT . ,gt n |n
v -~ - - - - -, c tanh (143)
dT [(!-) yx/c 2 ) 2 \ (/ 2 T 2 /c 2 Y- c.
on account of (140).
The velocity of the reference points relative to 1 thus depends on x,
and from the point of view of an observer in / the system of reference R
corresponding to the coordinates (x l ) will not appear as rigid The dis-
tance between two reference points (x, //, z) and (x-dx, y, z) measured by
an observer in / is found from (141) and (143).
j __ 7 __ //, -,2/ r 2\ ,/ r
T ~ C08h"to//C)^ " V( ^ ) '
(144)
where v ~ v(T) is the velocity relative to / of the system of reference R
at the place considered. From the point of view of an observer in /, each
part of the system R is thus contracting in accordance with the Lorentz
formula.
For small values of / and x, where w r e can neglect terms m gx/c 2 and
gt/c of higher than first order, the system of reference R considered here
is identical with the system considered at the end of 95.
VIII, 97 THE GENERAL THEORY OF RELATIVITY 257
Let us now consider the motion of a free particle in the gravitational
field of the system (x* ). The time track of such a particle is determined by
(86) with g lk given by (137), and for t = 1, 2, 3 (86) becomes
d 2 x _ /, , gx\ (dt\ 2
CiT \ C I \dT/
If the particle starts from a point on the #-axis with zero velocity, the
last equations (145) give __ _~
From (99) and (139) we get the following connexion between the
proper time of the particle and the time variable t of the system (x l ):
dr = dt^[(l+gx/c 2 ) 2 -u 2 lc 2 ], (146)
, dx
where u =
dt
is the velocity of the particle, (146) is equivalent to the last equation
(86).
Using t instead of r as independent variable in (145), we get by
means of (146)
d 2 x 2g/c 2 [dx\ 2 2 .
dt 2 l-{-gxlc 2 \dt)
The solution of this equation corresponding to the initial conditions
r r dx - -, __ M48\
X XQ, -= \J 1U1 & \J V 1 * /
dt
^ * = ?f( 1 + ? ^-ir-l'L < 149 )
as is easily verified by differentiation and substitution in (147).
The velocity of the particle at the time t is
dx - di I 03D
dt - + c*
Thus with increasing t the velocity increases, reaches a maximum
_ c(l+gx /c 2 )
a max S '
and decreases again to zero for t-+oo. If x > c 2 /g, the velocity of the
particle assumes values which are larger than c; however, u is always
smaller than the velocity of light, which is
w = c(l+gx/c 2 )
on account of (70) and (137).
3596.60 3
258 THE FOUNDATIONS OF VIII, 97
For t -> oo the particle approaches the singular wall x c 2 /g of our
system of coordinates. At this place also the velocity of light tends to
zero, and no signals of any kind will ever reach the boundary plane.
Using the expressions (149) and (150) for x and u in (146) we get
by integration for the proper time r of the particle at the coordinate
time t i
^o) f __<?!_ = /5 + ?o]tanh?-'. (151)
c*) ) coshV/c \g^ c) c V '
Finally, if the origin of the system (x 1 ) is moving with constant
velocity v in the direction of the positive X-axis,/, = 0, and the vector
</ t defined by ( 1 30) is zero. Thus the gravitational field is zero as it should
be, since the system of reference R in this case is a system of inertia
moving with the velocity v. Further, since U l is constant and/ t = U l r,
the transformation (127) is identical with the special Lorentz trans-
formation in this case.
98. The clock paradox
We are now in a position to state the complete solution of the clock
paradox which was mentioned in 20 and which played a certain part
in the early discussions on the consistency of the theory of relativity. f
Consider two standard clocks C l and (7 2 originally situated at rest at the
origin Oj of a system of inertia *S\ with the space-time coordinates
(X, 7, Z, T) (Fig 18). At the time T the clock Q is accelerated by a
constant force F in the direction of the positive Jf -axis. When C 2 has
reached the point A it has attained a certain velocity v, and from this
point on C 2 is allowed to continue m a uniform motion with the constant
velocity v until it reaches the point B, where it meets a constant counter-
force of the same magnitude F as before, but with opposite direction. C 2 is
brought to rest at C and accelerated back to J5, having then attained
the velocity v. Between B and A it moves with the constant velocity
v, and at A it is attacked again by the constant force F which brings
it to rest at O x . Let A'7 7 , A"2 T , A'" T be the times which C 2 takes to travel
the distances 0/1, AB, and BC, respectively. For symmetry reasons
the motion on the way back from G to A must be just the reverse of the
motion from A to (7, and further we must have
&"T - A'7 7 .
t A. Einstein, Ann d. Phya. 17, 891 (1905); P. Langevm, Scientia, 10, 31 (1911);
M. v. Laue, Phys. ZS. 13, 118 (1912), H. A Lorontz, Das Relativitatepnnzip, 3 Haar-
lemer Vorlosungen, pp. 31 and 47, Leipzig, 1914, A. Einstein, Naturiu 6, 697 (1918),
C. Mollor, Dan. Mat. Fys. Mldd. 20, No. 19 (1943)
VIII, 98 THE GENERAL THEORY OF RELATIVITY
259
Let 4r x and Ar 2 denote the measurements on the clocks C^ and C 2 of the
time elapsed between the two encounters of the clocks, T X and r 2 being
the proper times of C and <7 2 , respective^. Since C is constantly at rest
at the point 0, Ar t is equal to the total increase AT in the time variable
T of the system S 1 between the two encounters. Thus we have
A Tl - AT - 2(A'7 7 +A"T+A'"T) - 2(2A"F+A"T). (152)
y
^1
^ c ' ^ C 2
J V- )
% A ^ B C
^ , , , 7 , t
Via 18
Similarly we get
where r' 2 , rJJ, T'% denote the increase in proper time of C% during its travel
through OA, AB, and BC, respectively The motion of the clock C 2
from O to A is a hyperbolic motion and is thus described by the equation
(III.47),ic. * 2 , r /.J1V211 \
* = ! ll + F *-l, (164)
f7lL \c / J I
^
where
(155)
(156)
(157)
and r/? is the rest mass of 6V The velocity u = dX/dT is thus
dX gT
u
Hence we get
v
or
260 THE FOUNDATIONS OF VIII, 98
Using (155) we can now calculate the proper time r' 2 of C z by means of
the formula (II. 38) of the special theory of relativity valid in the system
of inertia S v Hence
AT AT
On account of (157) this may also be written
q&T vie
. , o . , /lr o\
= Slnh = Sinh - ( 158 )
c c
or tanh = - - - - - = . (1 58')
./(n-
In the same way we get from (II. 38)
v 2 / c2 )- ( 159 )
Now, if for constant value of v we apply a larger and larger force
F the acceleration g = Fjm will also increase. In the limit g -> oo for
constant v we see from (158) that both AT = tt"T and r' 2 r,J tend
to zero. In this limit, where the velocity v is attained nearly instan-
taneously, we thus get from (152), (153), and (159)
A Tl - 2A // 7 7 , Ar 2 = 2rl = 2&"TJ(l-v*[c 2 ), (160)
i.e. Ar 2 - Ar x V(l-^ 2 /c 2 ) (161)
as we should have. The moving clock (7 2 is lagging behind the stationary
clock C v Further, in the limit g -> oo, the maximum distance Z between
the two clocks is simply given by
J = vAT. (162)
We shall now see that the same result is obtained if the whole process
is treated in a rigid system of reference *S 2 with coordinates (x,y, z, t)
which follows the motion of C* 2 in such a way that C 2 is permanently
situated at the origin. In the time intervals where S 2 is accelerated rela-
tive to J&! or to the distant stars, we have a gravitational field in 5 2 .
During the time interval < t <. r 2 of magnitude A' = r 2 , the gravita-
tional field is described by the scalar potential (139). In the interval
T' Z < t < T2+ T 2 f magnitude A"/ r 2 , we have x an( i i n ^he
interval ri+Vj < / < r^+rl+r^ of magnitude A w == r 2 " = r 2 = A'^ we
have x ^ {7^(1 (7#/2c 2 ). During the first period A'tf the clock Cj is
falling freely in the direction of the negative x-axis in accordance with
VIII, 98 THE GENERAL THEORY OF RELATIVITY 261
the equation of motion (149). In the period A" it is moving uniformly
with the velocity -~v and, finally, in the period A'" it is brought to rest
at a point with the coordinates a* = 1. Since the systems S l and S 2
are at rest relative to each other at that moment, the maximum distance
between the two clocks is the same when measured in S 2 or in S lf After this
moment the clock C returns to the origin with reversed motion,
The clock C* 2 is at rest at the origin of S 2 during the whole process, since
the gravitational field is counterbalanced by the external force F.
The increase in the proper time of the clock C l can now be calculated
by means of the general formula (99) with the expressions for x given
above. In (149), (150), and (151) we have given the solution of the
equations of motion (147). If r\, TJ, and r denote the increase of the
proper time of C l in the intervals A', A '7, A "7, respectively, we have
obviously An = 2( T ;+rI+TD. (163)
Similarly, since C 2 is at rest at the origin x = 0, where x * s constantly
equal to zero, we have
AT, = 2(A'J + A"H-A'"0 - 2(2A'H-A"/) - 2(2r' 2 -f r 2 '). (164)
Since C\ starts from the origin with zero velocity, we can get r[ from
(151) by putting X Q = and / = A 7 = r 2 , Hence
- (165)
c 9
on account of (158').
During the interval A% the clock (7 X is moving with constant velocity
in a field-free space. Thus we get
T\ -~ A"/V(1~W 2 ) - r^(l -v*Ic 2 ) = A"T(l-v 2 /c 2 ) (166)
on account of (159). Finally, r may be obtained by putting y equal to
-(/, XQ = -/, and t = A"7 - r ^ r' 2 in (151). Hence
rr = (MtaBh^ = (MH (167)
1 \g^c) c \f7 c/c
on account of (158').
In the limit as g ~-> oo with constant v we get from (165) r\ -> 0. But
although A'" r 2 " = r 2 -> in this limit, rj' approaches the finite limit
This surprising result is due to the influence on the rate of the clock C l
of the gravitational scalar potential x = $ x ( 1 J7#/2c 2 ) which becomes
infinite in the limit g->ao.
262 THE FOUNDATIONS OF VIII, 98
Thus, in the limit g -> oo, we get from (163), (166), (168), (162), (164),
and (159),
, (169)
Ar - 2r - 2
i.e. the same result as in (160). This result which represents the solution
of the clock paradox is, of course, not surprising, since the proper time is
an invariant which has the same value in any system of coordinates.
We shall now finally consider another much simpler example of the
same phenomenon in which also the importance of the gravitational
vector potential for the rate of moving clocks is illustrated. Consider a
clock (7 2 which under the influence of a central force F performs a uniform
circular motion in a system of inertia AS\. If the radius of the circle is R
and the constant angular velocity o> p the velocity of the particle is Rw^
and the increase in the proper time r 2 during a revolution is
r 2 = 7V(l-# 2 to 2 /c 2 ) - J(l-R*a>*lc*) (170)
o>
according to the formula (II. ?J8) The corresponding increase in the
proper time of a clock C\ at rest at the periphery of the circle is
27T
r
l = T = ~". (171)
CO
Let us now treat the same phenomenon from the point of view of an
observer on the rotating disk of 90. In this system # 2 the clock (7 2 is at
rest at the point (r R, & ~~ 0), say, while the clock C l is rotating with
the angular velocity yn
'
in a circle of radius / It. (\ is falling freely under the influence of the
gravitational field with the potentials (74) and (97),
(in)
It is easily seen that > constant, d&/dt = w is a solution of the
equations of motion (86) with g tfc given by (73). The clock C 2 remains
at rest, the gravitational acceleration (96) being counter-balanced by the
force F.
VIII, 98 THE GENERAL THEORY OF RELATIVITY 263
For the increase r 2 of the proper time of C 2 during the time t ZTTJOJ
we now get at once from the formula (100) for a clock at rest
r 2 = VU + 2*/c 2 ) = Va-r^/c*) (173)
CO CO
in accordance with (170).
To determine the corresponding increase of the proper time of C\ we
have to use the general formula (98). Since
we get from (74)
da 2 dx l dx l
Further, using (172),
Hence, from (98),
TI = 27r f(V(l---r g aj 2 /c a ) | ***** V ***** j^ = 27T -
(174)
Thus, in this case, the effects of the gravitational potentials and of the
velocity of the clock C\ on the proper time r l as expressed in the formula
(98) just cancel and for TJ and r 2 we get the same values as before.
The physical interpretation of the effect is, however, completely dif-
ferent in the two cases Tn S 1 the effect is ascribed to the velocity of
the particles only, while in S 2 the phenomenon is explained as a joint
effect of the gravitational field and the motion.
IX
PERMANENT GRAVITATIONAL FIELDS.
TENSOR CALCULUS IN A GENERAL
RIEMANNIAN SPACE
99. Four-dimensional formulation of the general principle of
relativity and of the principle of equivalence
IN the preceding chapter we have considered the case of gravitational
fields which could be transformed away by the introduction of the
pseudo- Cartesian coordinates of the system of inertia / from which we
started in 88. We have seen that the action of the gravitational field
in an arbitrary system of coordinates (x l ) is described by the metric
tensor g lk which determines the line element in space-time by the
ec * uation *
(1)
The non-permanent gravitational fields are thus characterized by the
property that the interval can be brought into the form (VIII. 40)
ds* - G lk dX*dX k (2)
for all points in 4-space by a suitable choice of the space-time coordinates
or, in other words, the space is a flat pseudo-Euclidean space in this case.
According to the principle of equivalence there should, however, be
no essential difference between permanent and non-permanent fields,
both types of fields satisfying the same fundamental laws. We shall
therefore assume that the gravitational fields produced by the presence
of large masses as, for instance, that of the earth or the sun, are described
by the metric tensor y tk in 4-space in the same way as in the case of the
artificially produced non-permanent fields. In particular, it is assumed
that the time tracks of a free (i.e. freely falling) particle and of a light ray
traversing permanent gravitational fields are geodesic lines in 4-space,
given by the same equations (VIII. 86) and (VIII. 87) as in the case of
non-permanent fields. The only difference will then be that the per-
manent fields cannot be removed completely by a suitable transforma-
tion of the space-time coordinates, i.e. ds 2 cannot be brought into the
form (2) simultaneously for all points in 4-space. Hence, in this case, the
4-space is a 'curved' space with a general Riemannian geometry.
As we shall see in 104, it is, however, always possible in an infinite
number of ways to introduce a so-called geodesic system of coordinates
for which the first derivatives d lk ldS? of the metric tensor are zero
IX, 99 PERMANENT GRAVITATIONAL FIELDS 266
and the values of lfc are equal to O tk at a given point in 4-space. Geo-
metrically this means that the space may be treated as flat in an infinitesi-
mal region around every point, in analogy to the two-dimensional case
where a curved surface may be replaced by the tangential plane in a small
region around the point considered. Systems of space -time coordinates
(#') with the above-mentioned properties may also be called 'local
systems of inertia* ; for in the case of permanent gravitational fields they
play locally the same role as the systems of inertia in the case of non-
permanent fields. For an arbitrary choice of space-time coordinates
(x l ), i.e. for an arbitrary numbering of the events in physical space,
the quantities g lk can therefore always be experimentally determined
by the same methods as those described in 89.
According to the general principle of relativity, the laws of nature
must now be expressible in the form of equations which are form-
invariant. " Thus, if the law in question is expressed by equations of the
B - dA dB }
' B '""''"' =0
where A y B,... are physical quantities, we have in another arbitrary
system of coordinates (x >l ) the same functional relation between the
physical quantities in (x' l ) y i.e.
The only difference from the case considered in 35 is that now the gravi-
tational quantities g lk will be among the set of physical quantities
A, J5,... entering into the equations (3).
In the special theory of relativity, the covariance of the laws of nature
under Lorentz transformations could be very elegantly expressed by
means of the four-dimensional tensor calculus. To obtain a similar
representation of the laws of nature in the general theory of relativity
we shall have to make a generalization of the tensor calculus developed
in Chapter IV for pseudo- Cartesian systems of coordinates.
In the case of non-permanent gravitational fields, this generalization
simply consists in the rather trivial extension of the notion of vectors
and tensors to general curvilinear space-time coordinates, the geometry
of 4-space being the same as in the special theory of relativity. In the
general case of permanent fields, however, the structure of the 4-space
itself is different and the development of a tensor calculus for a general
Riemannian space will be required. Formally the difference between
these two cases is not very great and, as we shall see, most of the tensor
266 PERMANENT GRAVITATIONAL FIELDS IX, 99
relations holding for curvilinear coordinates in a flat space can be used
also in a general curved space.
100. Contravariant and covariant components of a four -vector
Let (x l ) be an arbitrary system of curvilinear coordinates in 4-space.
The geometry in this space is then completely defined if the components
fj ik of the metric tensor are given functions of the space-time coordinates.
We shall assume that the (j lk satisfy the conditions (VIII. 52) at every
event point This means that the determinant
011 012 013 014
== I0i*l =
(4)
031 032 033 034
041 042 043 044
is negative. Let A lk be the conjugate minor of the element g lk in the ith
row and kih column. From well-known theorems of the theory of
determinants we then get
(no summation over i) \. (5)
and
i
Defining now a symmetrical scheme of quantities g tk by
g* - d* = / S (6)
we have, according to (5),
0,/ff* 1 = ?* = 8?, (7)
where 8* = ( l f f i = * (8)
I for i -^ k
If we introduce a new system of coordinates Jby the transformation
x' 1 =- r''(j<?) (9)
we get, exactly as in the case of a flat space considered in 88 (see equa-
tions VIII. 54-58),
' 1 - 4 dx" = ~ dx"
,
= a' k dx' k = dx' k
(11)
g\ k = &\s$g lm I
9tk = <*(<*kSi m I
IX, 100 PERMANENT GRAVITATIONAL FIELDS 267
In virtue of the transformations (10), (12) the expression for the interval
is invariant ^ _ ^ ^^ = ^ ^ w *. ( 13)
It is always possible to choose the transformation (9) so as to make
g lk equal to the G lk defined by (VIII. 41) at a given point in 4-space. In
general it will, however, not be possible tp obtain this result simul-
taneously for all points; for the coefficients aj, cannot be chosen freely
since they must satisfy the integrability conditions
following from the definition of the aj. in (10). This will be possible only
for a flat space, in which case the functions g lk satisfy special conditions
(see 107).
A vector at a definite point (x l ) is now defined as a quantity which has
four components a 1 in every system of coordinates satisfying the same
transformation law as the coordinate differentials in (10), i.e.
a' 1 ==- 4a fc . (15)
On account of (11) the reciprocal relations are
a 1 = & k a' k . (15')
Since the coefficients a k and & k are functions of the coordinates (a: 1 ),
the transformation laws (15) and (15') have a definite meaning only
with reference to the particular point with which the vector a 1 is con-
nected.
In curvilinear systems of coordinates we have to distinguish between
the contravanant components with the transformation equations ( 1 5) and
the covanant components a l of the same vector, defined by the equa-
in every system of coordinates. On account of (7) the relations reciprocal
to (16) are *__* m\
a 9 a k- ( l{ )
The operations (16) and (17) are called lowering and raising of indices.
In a Cartesian system of coordinates in Euclidean space we have
and there will be no difference between the covariant and the contra-
variant components.
268 PERMANENT GRAVITATIONAL FIELDS IX, 100
From (12), (15), and (11) we get the following transformation equa-
tions for the co variant components:
or
the reciprocals of which are
In a flat (3 -fl) -space we can use pseudo-Cartesian coordinates and
the connexion between the covariant and the contravariant com-
ponents of a \ector in this system of coordinates is simply
or a 1 ~ ,
i.e. in this case we have O tk = # ?fc (20')
in accordance with (7) and (VIII. 41). If we had used the real representa-
tion of vectors in the special theory of relativity, it would have been
necessary to distinguish between covariant and contravariant com-
ponents in Chapter IV. It was just in order to avoid this slight com-
plication that the imaginary time components were introduced by
which the 4-space was made fornially Euclidean. In any case, it is easy
to pass over from the imaginary to the real representation. It is only to
be remembered that the contravariant components of a vector in the
real representation are obtained from the components in the imaginary
representation by dropping the symbol i in the fourth component. The
covariant components of the vector are then obtained from (20).
From this rule it follows that the norm of a vector, which in the
imaginary representation was defined by (IV. 25), in the real representa-
tion takes the form 2 __ i /oi\
and, since this expression is invariant for all coordinate transforma-
tions, (21) will hold also for curvilinear coordinates. In fact, we get
from (18), (15), and (11)
a\ a' 1 = dt{a; ft 0,a m = Sj rt a,a m = a, a'- ( 22 )
We may therefore take (21) as the norm of a vector also in a general
Biemannian space. According to ( 16) and (17), (21 ) may be written in the
different forms ^ = ^ _ ^^ = ^^ (23)
Similarly, the vector product of two vectors a and b is given by the
invariant
IX, 101 PERMANENT GRAVITATIONAL FIELDS 269
101. Tensor algebra
The generalization of the tensor calculus developed in 39-44 for
Cartesian systems of coordinates to the general curvilinear coordinates
of Riemannian space is now obvious. A tensor of rank n in 4-space is a
quantity with 4 n components which transform with respect to each
index like a vector, i.e. the transformation is given by (15) or (18),
according as the tensor is contravariant or covariant with respect to
the index in question. The connexions between the covariant and
contravariant components of a tensor are given by the general rules
(16) and (17) for lowering and raising indices.
The transformation laws for the contravariant and covariant compo-
nents of a tensor of rank 2 are thus
*"* = ajc&#* (25)
4 = &\&Wi m > (26)
respectively, and the connexions between the covariant and contra-
variant components are
(27)
'in every system of coordinates. On account of (11) and (12), the equa-
tions (25-27) are easily seen to be compatible. Besides the purely
contravariant and covariant Components we can also form the mixed
components
tf = 0% = 0,i* and t\^gt ik ^g ki v l > (28)
which transform according to the laws
In general, t\ will be different from t k l .
All symmetry properties like
iik __ _^ t ki (30)
are invariant properties. They may also be expressed as
which, on account of (27), (28), are equivalent to (30). Thus the mixed
components t* and t k i of a symmetric tensor are equal and may therefore
simply be written as t k .
A comparison of (12) and (26) shows that the quantities g ik themselves
are the covariant components of a symmetrical tensor of rank 2 the
metric tensor. Further, we see from (7) that the mixed components g* of
270 PERMANENT GRAVITATIONAL FIELDS IX, 101
this tensor are given by the Kronecker symbol 8f and that the contra-
variant components of the metric tensor are equal to the quantities
g lk defined by (6).
Similarly, as in the case of Cartesian coordinates considered in 41,
we can now also in the general case form new tensors by the processes
of addition, direct multipli cation, and contraction. By addition of two
tensors of rank n we get a new tensor of rank n, and by direct multiplica-
tion of two tensors of the ranks n and m we get a tensor of rank n-\-m.
It should be remarked, however, that these processes have an unam-
biguous meaning in the general case only if the two tensors belong to
the same point in 4-space. Finally, the process of contraction which in
curvilinear systems of coordinates consists in equating an upper and a
lower index, and summing, reduces the rank of a tensor by 2. By con-
traction of a tensor of rank 2 we thus get a tensor of rank 0, i.e. an
invariant. From the transformation law (29) we see at once that
is an invariant which, by means of (28), may be written in the different
forms / * /T *M n^t f k (w\
l i ( Jik l (/ l ik l k- V)
An example of a combined application of the processes of direct
multiplication and contraction is offered by the equation (24).
102. Pseudo- tensors. Dual tensors
Let a = | af | and a = |a* | be the determinants corresponding to the
scheme of transformation coefficients cxf and af , respectively. According
to the multiplication rules for determinants, we get at once from (11)
.=!, (34)
and from (12) g' = \g( k \ = dc.g.ci = & 2 g = ---,
where |a| and |a| are the absolute values of the determinants a and a.
A pseudo-tensor is now defined as a quantity whose components
transform like the components of a tensor, except that they are multiplied
by the sign = of the transformation determinants. If ex and
|a| |a|
consequently a are positive, a pseudo-tensor therefore transforms like
a tensor of the same rank*. In the same way as in 43, it is now easily seen
IX, 102 PERMANENT GRAVITATIONAL FIELDS 271
that a quantity A lktm , whose components in every system of coordinates
are equal to the Levi-Civita symbol 8 lWm , transforms according to the law
A' tklm = otflSt&lc&A^. (36)
From (35) and (36) we thus see that the quantities
are the covariant components of a completely antisymmetric pseudo-
tensor of rank 4.
In the same way as in 44, we can now to an antisymmetrical tensor
of rank n adjoin a dual pseudo-tensor of rank 4 n by means of the
pseudo-tensor (37). Thus, if F lk are the contravariant components of an
antisymmetrical tensor, the covariant components of the dual tensor
are given by ^ ^ ^ F , m = y ( _ g) ^ ^ (sg)
i.e.
" 1 3t (39)
'
Two infinitesimal vectors a 1 , b k define a parallelogram described by
an antisymmetrical tensor with the contravariant components
O ik ^ a l b k a k b l , (40)
the area a being given by or 2 = \o lk a lk .
The corresponding dual tensor
is orthogonal to a ?Ar , i.e.
Three infinitesimal vectors a 1 , b k , d define a three-dimensional parallele-
piped described by the antisymmetrical tensor
a 1 b l c'
k b k c k
' V c l
or by its dual pseudo-vector
(42)
(43)
which is orthogonal to the vectors a\ b l , c l , i.e.
^a l == V % b l = V,c l = 0.
The volume V of the parallelepiped is given by
p=_F l P=-ii ftll F*. (44)
272
PERMANENT GRAVITATIONAL FIELDS
IX, 102
Finally, four infinitesimal vectors a\ b k , c l > d m define a four-dimen-
sional parallelepiped described by the tensor
;< V C l d*
c k d k
c l d*
c m flm
a*
a 1
a m
b m
(45)
or by the dual pseudo-invariant
V * V.Wm _ V(-flO'
4!
4'
.e.
V(-P)
a
a
= v<-
d 3
d 4
(46)
Jf a z ', 6 1 , c l , d l are infinitesimal vectors lying in the directions of the
coordinate curves and of lengths dx l , dx 2 , dot?, dx*, respectively, we have
a 1 = (da: 1 , 0,0,0), b l = (0,dx 2 , 0, 0), etc., and the corresponding four-
dimensional volume element is given by the pseudo-invariant
dZ = V(-flO dx*dx*dx?<fa*. (47)
This is the generalization of the expression (VIII. 17) for the volume
element in a two-dimensional space with positive definite metric. In a
3-space we can similarly adjoin a pseudo-tensor of rank 3 n to an
antisymmetrical tensor of rank n. (See Appendix 5.)
103. Geodesic lines. Christoffel's formulae
The geodesic lines are defined by the equations (VIII. 30)
dX ik ~dX
-
gik ~ ~~
dA
(48)
where A is an arbitrary invariant parameter and the indices i, k, I run
from 1 to 4. (48) may also be written
ik l fyu\ dxk d ^ _,
Multiplying this equation by g mi we get, on account of (7),
'_^W\^ ^ _
*
or
(49)
IX, 103 PERMANENT GRAVITATIONAL FIELDS 273
with
The quantities Fjj, T lkl defined by (50) are the Christoffel three-index
symbols. They obviously satisfy the relations
F l = F' F F --^ ~ F 4-F (51)
The connexion between F^ and F t w is the same as that between the co-
variant and contravariant components of a tensor but, as we shall see,
the Christoffel symbols do not transform like a tensor.
Since the equations (48) or (49) are the Euler equations corresponding
to the invariant variational principle (VIII. 21, 29), they must hold in
every system of coordinates. By differentiation of the equations (10)
we therefore get
dx l _ w| dx' r
dX ~ ' ^ d\
U> X ^ (JL X GOL^ CLX (J/X /r\^\
ic,' 1 } dX dX
where we have used the equations (49) in the system of coordinates (x' 1 )
Using (52) in (49) we get
^ -'diA-- ==0
and, since this equation must hold for independent values of the variables
dx' k /dX, the expression inside the brackets, which is symmetrical in k and
I, must be zero. If we multiply the relation obtained in this way by a
we get, on account of (11), the Christoffel formulae
(53)
By linear (affine) transformations, where the coefficients d, a are
constant, the first term on the right-hand side of (53) is zero and the
Christoffel symbols transform like the components of a tensor. For more
general transformations this will not be the case, and the Fj^, T ikl are
therefore called affine tensors.
If the g lk are constant, as in the case of the pseudo- Cartesian system of
coordinates in a flat space, the Christoffel symbols (50) vanish.
3695.60 T
274 PERMANENT GRAVITATIONAL FIELDS IX, 104
104. Local systems of inertia
In general, it is not possible to introduce a system of coordinates which
makes the components of the metric tensor independent of the co-
ordinates, but, as we shall see now, we can always ensure that this is
approximately true in the immediate surroundings of a given point P in
4-space. More exactly, we can always find a system of coordinates ($J
a geodesic system for which ()g lk /dx l ~ at the point P. Let (x l ) be the
original system of coordinates, and let T' kl (P) be the values of the
('hnstoffel symbols at the point P. If (x' P ) denote the coordinates of
this point, the transformation
x' x l -x^+\r^(P)(* r -~x r l >)(x*--~x 8 1 ,) (54)
will lead to the desired result. If we identify the primed system in (53) with
the system (x l ) we get from (54)
At the point P we thus have
&UP) -- i, ?| (P) = - r;,(m &;, (55)
Cj X
and if we use (55) in (53) we get
l"'u(P) = (56)
at the point P. From (50) and (51) we then find that also T ltkl and
d i jdx l are zero at this point. Further, the components of the metric
tensor y lk at P are, on account of (55),
and the derivatives of the gravitational potentials y l and % defined by
(VIII. 63, 94) are zero. Thus the gravitational acceleration is zero at P,
the gravitational field has been locally transformed away. The system
of reference R corresponding to the coordinates x l is a local system of
inertia.
o o
The motion of the origin O of the system R relative to the original
system R with coordinates (x 1 ) is obtained from (54) by putting & = 0,
which gives
- jr*-a+
IX, 104 PERMANENT GRAVITATIONAL FIELDS 275
By differentiation of this equation with respect to an arbitrary parameter
A we get
o _ , r , m-
dw + rs( ' dX~ (
At the time t = t p = Zp/c, corresponding to the event P, the space-time
coordinates of O are
x l = x'r and x l = 0.
O ,
At this moment the motion of O is thus given by the equations
1 2l r r tJjr 8
- i) - (58)
which are the equations of motion of a freely falling particle v/hich is
momentarily at rest relative to H (cf. (49)).
Any geodesic line which goes through the point jP, including the time
track of freely falling particles and light rays, is in the system # l described
by the equations (49) with the Christoffel symbol given by (56) Hence
we get simply o t
d * = 0. (59)
dX 2
In a small region around P the local system of inertia thus has the same
properties as a usual system of inertia. Without any change in the
system of reference we may now further introduce local pseudo- Cartesian
coordinates X 1 m R, by a transformation of the type (VIIT. 59), so that
the metric tensor G lk in this new system is
(1 for i = fc=- 1,2,3
- 1 for i = k - 4 (60)
for i -- k
at the point P. The necessary transformation is even linear, therefore
the derivatives of the metric tensor will remain zero at P, i.e.
o
8 ' k (P) = 0. (61)
dx-
In the following we shall always use such space-time coordinates in
the local systems of inertia. By Lorentz transformations of the variables
X 1 we can pass over to new local systems of inertia which in general
will be moving relative to the original one.
276 PERMANENT GRAVITATIONAL FIELDS IX, 104
o
In a small region around P, where terms of second order in X 1 can be
neglected, the metric tensor may be regarded as constant in these
systems. In accordance with the principle of equivalence, it is now
assumed that all laws of nature at the point P have the same form as in the
special theory of relativity when expressed in terms of the local pseudo-
Cartesian coordinates X 1 . By a simple transformation of coordinates
we may then obtain these laws in a general covariant form. This requires
the development of tensor analysis in a general Riemannian space.
105. Parallel displacement of vectors
Let a 1 be the contravariant components of a vector at the point (x l ).
By means of the Christotfel formulae it is now easily seen that the
quantities a*' _ a l -\~d if a l with
d t) a l = -r kl dx k a l (62)
transform like the contravariant components of a vector at the neigh-
bouring point (x* -\-djr 1 ). From (53), (10), and (11) we obtain
(63)
4+^ = (in
k l dx"> v }
Further, we get by differentiation of (11)
?*! 4+^ =
ox n k l dx">
and the first term in the brackets may therefore be written
where we have made use of (1 1) and (14).-
Hence (63) becomes
Pa*
d n a' 1 = ^- '- l dx m a n +ot l r da r ,
J> Q x m \ r p >
and if we neglect terms of second order in dx l the transformation law for
the quantities a 1 +rf^a l can be written
= * l H (x+dx)(a*+d p a) 9 (64)
where <x,' n (x-{~dx) are the transformation coefficients taken at the point
(x 1 +dx l ). (64) shows that the quantities a* 1 = a l +d p a i are the contra-
variant components of a vector at the point (x^+dx 1 ). In a flat space,
IX, 105 PERMANENT GRAVITATIONAL FIELDS 277
the vector a* 1 is identical with the vector obtained by parallel displace-
ment of the vector a 1 from the point (x l ) to the neighbouring point
(x t -\-dx i ) ; for if we introduce pseudo- Cartesian coordinates, the Christoff el
symbols and therefore also d p a 1 vanish and the components a* 1 and a 1
of the two vectors are equal in this system. If we use a curvilinear
system of coordinates in a flat space, the contravariant components of
the two vectors will, thus, differ by the amount d lt a l defined by (62).
It is now natural also in a general Riemannian space to define the
parallel displacement of a vector by the equation (62).
If X i is a local system of inertia for the point considered, the com-
ponents a 1 of a vector at this point are unchanged by parallel displace-
ments exactly as in a pseudo- Cartesian system of coordinates in a
pseudo-Euclidean space.
From (62) it follows that the norm of a vector, and also the scalar
product of two vectors a 1 and b l at the point (x l ), are unchanged by a
parallel displacement, for from (62) we get
(65)
on account of the identity
^f-flV^-^rj^O (66)
following from (50) and (51).
Using one of the other forms (24) in which the scalar product can be
written, we get
= d p (a t b l ) = d p a l .b l a l Y l kl dx k b l
Since this equation must hold for an arbitrary vector 6', we get for the
change in the covariant components of a vector by a parallel displace-
ment d,a t = TU dx*a, = r,. lfc dxW. (67)
Finally, expressing the scalar product in the form g lk a l b kt we get by
means of (67)
= <7'*a)
278 PERMANENT GRAVITATIONAL FIELDS IX, 105
and, since this equation must hold for arbitrary dx\ a iy 6 t , the equation
0~1 If
\ == (68)
must be identically satisfied.
Multiplying (66) by g lk and applying (7), we obtain
Farther we get by means of a well-known theorem the derivative of the
determinant </ - |j/ (fr | in the form
where we have made use of (6) and (7).
Hence r- 11 ^ 1 ' (69)
a relation which we shall use later.
Consider again a geodesic line defined by the equation (49). If A is
an invariant parameter, , .
is a four-vector lying in the direction of the tangent. The equation
(49) may then be written , 1Jl
A comparison of (71) with (62) shows that the different vectors U l along
the geodesic line are obtained by parallel displacement along this line, a
property which they have in common with the straight lines in a Euclid-
ean space. The geodesic line connecting two points is thus not only the
line with a stationary value of length, it is also the 'straightest' line.
The norm of the vector U l must therefore be constant along the line,
i.e. g^ k U l U k is independent of A, in accordance with (VIII. 31).
If this property of the geodesic lines is expressed in terms of the
covariant components t^ of the vector dx l /dX, we get, according to
(67) and (51),
*% = r llk uw = Kr w +r M )iw' = ~ ^ v k u*. (72)
This equation is identical with the equation (48).
The equations (62) define the change in the components a 1 ' of a vector
by an infinitesimal parallel displacement along the vector (dx l ). The
IX, 105 PERMANENT GRAVITATIONAL FIELDS 279
total change of a 1 by a displacement along a finite curve may thus be
obtained by integration. In aflat space the total change of a 1 by parallel
displacement along a closed curve will be zero. This is seen at once if
we use a Cartesian or pseudo- Cartesian system of coordinates, for in
this system the components a 1 are not changed at all by the displace-
ment. The final vector a* 7 obtained by the displacement along the
closed curve is thus equal to the original vector a 1 , and this must then
also be true if we afterwards introduce curvilinear coordinates. In a
curved space, however, the final vector a* 1 will in general be different
from a 1 , the difference a* l ~a l depending on the closed curve (see 107).
Thus, if a vector is given a parallel displacement from a point P l to a
point 1 g along a certain curve connecting I\ and P 2 , the resulting vector
a* 1 will depend on the form of this curve if the space is curved, while it is
independent of the curve in a flat space. This is in fact the only essential
difference between curved and flat space.
106. Tensor analysis. Covariant differentiation
As in the case of the flat pseudo-Euclidean space of the special theory
of relativity, we speak of a tensor (or pseudo-tensor) field of rank n in a
general Ricmannian space if a tensor (or pseudo-tensor) of rank n is
connected with every point in this space. Similarly as in 48, we can
derive from such a field a new tensor field of rank n-\- 1 by a differentia-
tion process. From a tensor field of rank 0, i.e. from a scalar field
f (*') = ^(x), (73)
we can thus derive a vector field grad </> with the co variant components
grad,* = g. (74)
By means of (73) and (10) we get at once
which shows that the d<f)/dx L are in fact the covariant components of a
vector.
However, if we try in the same way to form a tensor field of rank 2
from a vector field a 1 by differentiation of the transformation equations
,76,
v '
From this equation we see that the da l /3x k will be the mixed components
of a tensor only if the coefficients aj are constant. On the other hand,
280 PERMANENT GRAVITATIONAL FIELDS IX, 106
we get by means of the Christoffel formulae (53) and the relations ( 1 1 ), ( 14) ,
and (IT) (p. 276)
(77)
By addition of (76) and (77) we find that the quantities
a'^g+IlX (78)
transform according to the law
Thus, by the process of 'covariant differentiation' (78) of the contra-
variant components a 1 of a vector w r e get the mixed components of a
tensor field of rank 2.
This process may be described geometrically in the following way.
Let a l (P) and a l (P') be the vectors of the vector field connected with
the neighbouring points P and P' with coordinates (x 1 ) and (x l -\-dx l ),
respectively. The differences
da 1 - a*(P')-a*(P) - f^* dx k
will then not be the components of a vector, since a'(P) and a'(P')
belong to different points. However, if a*'(P') denotes the vector
obtained by parallel displacement of a 1 from P to P', the difference
a l (P')-a* l (P') = a l (P f )~c
will represent an infinitesimal vector at the point P'. Neglecting terms
of the second order in (dx k ), a l ^ k dd* will also be a vector at P, and since
this must hold for arbitrary infinitesimal vectors dx 1 , the quantities a\ k
must be the mixed components of a tensor of rank 2.
If this consideration is applied to the covariant components a t of the
vector field, we find, using the formulae (67) instead of (62) for the
parallel displacement, that the quantities
a '- fc "~ r ' jia ' (79)
are the covariant components of a tensor. Tn a geodesic system and, in
particular, in a Cartesian system of coordinates in a flat space, where
the Christoffel symbols are zero, the covariant differentiations (78) and
(79) are ordinary differentiations.
IX, 106 PERMANENT CSRA\ JTATIONAL FIELDS 281
Let us now consider the special case, where we have only a vector
a l (A) connected with each point on a cm ve with the parametric representa-
tion z*= * '(A).
We can then define the covariant derivative of this vector with respect to
rfA
The quantity _Da*(A)/dA is obviously a vector at the point P with the
coordinates #'(A); for from (62) we see that
Do' . [o'(f')--- _ ,
- = llin - - ----- - ; Jllll ,
d\ P'-+P AA i"-*p AA
(80')
where P ; with the coordinates a; l (A+AA) in this limiting process is
approaching the point P along the curve. Similarly we get for the co-
variant derivative of the covariant components a t
''-'
By means of (51) we get
i.e. DaJdX and Da'jdX are the covariant and contravariant components
of the same vector.
According to (71) and (72) the equations for the time track of a free
particle can now also be written
= or _
d\ d\
The process of covariant differentiation may be applied also to
tensor fields of higher rank. ( Consider, for instance, a tensor field of rank
2 with contravariant components V k . Since each index is transformed
282 PERMANENT GRAVITATIONAL FIELDS IX, 106
separately according to the same rule as for a vector, it is clear that the
quantities ^. tfc
with two contravanant indices and one covariant index, are the mixed
components of a tensor of rank 3. This may be verified in the same way
as for a vector field by means of the transformation laws of tensors and
Christoffel symbols. Similarly it is seen that
/ ( t>k pr t pr 4
l *W ~ T-J l <MrA, A AM<r
^ ; (82)
f/< k L ]i) ir iv /
-TTj 'T V lr t k~ i kl t 'r
are purely covariant and mixed components, respectively, of a tensor
of rank 3.
These rules may be extended to the case of a tensor field of rank n,
the number of terms containing Christoffel symbols being n in this case,
and if we put ,
< t - -* (83)
Ytl dx
in the case of a scalar field, this rule also applies to tensor fields of rank 0.
Since the scalar product aj) 1 of two vector fields a 1 , b l is a scalar field,
we have
'.*, (M)
i.e. the usual rule for differentiation of a product holds also for covariant
differentiation. This rule is easily seen to hold for the contracted
product of any two tensors of arbitrary rank ; for instance we have
(t lk a k ) tl = t lk tl a k +V k a kil )
The identities (66) and (68) may now be written
g lkjt - 0, g*j ~ 0, (86)
i.e. the covariant derivatives of the metric tensor are zero. By covariant
differentiation of (16) and (17) we therefore get
"i,* = 9ik,i<*> k +g lk a k tl - g lk a\i, a', - g*a kjt , (87)
i.e. the quantities a lk and a l fk are components of the same tensor of
rank 2. In the same way we see that the quantities t ik j, P k j, and t lkl>
defined by (81) and (82), are components of the same tensor of rank 3.
IX, 106 PERMANENT GRAVITATIONAL FIELDS 283
To obtain a generalization of the differential operators defined in
48 for the special case of pseudo-Euclidean space, we simply have to
replace the differentiations in 48 by covariant differentiations. For
the curl of a vector field a l we thus get
since the terms containing the Christoffel symbols cancel.
The covariant expression for the divergence of* a vector is obtained
by contraction of the tensor a 1 fc , i.e.
}==a< fl = +r< r ar. (89)
On account of (69) this may also be written
** = s+JBi s> - 561 www ""
In the (3+1) -space (/ is negative, i.e. \g\ </, for a space with positive
definite metric tensor g > and \g\ = g. The covariant expression for
d'Alembert's operator is now
(91)
The contra variant components of the divergence of a tensor field T tk are
K ' (92a)
and similarly we get for the covariant components
div ( {T t *} = T** = ^L^(\g\)Tf)-r r , ta T'<. (926)
For a symmetrical tensor this reduces to
T " (92c)
on account of (51).
For an antisymmetrical tensor F lk , the last term in (92 a) is zero on
account of the symmetry of Fj^ in the lower indices; hence
(93)
284 PERMANENT GRAVITATIONAL FIELDS IX, 106
Further, we get for the curl of an antisymmetrical tensor
l (94)
since the terms containing the Christoffel symbols cancel.
The generalization of Gauss's theorem (IV. 191) is obvious. Using
(43), (47), and (90), we get for an arbitrary vector field a 1 in the (3 + 1)-
space
f div{a} dZ --= f ,f t (V(-(7)a l )
L
- f a' dV> =- J a'V(-ff) 8 ljHwl rf^S^A^, (95)
r
where (dx k ), (8xf), (A# WI ) are three infinitesimal vectors in the three-
dimensional boundary V of the four-dimensional region.
107. The curvature tensor
Let a k be an arbitrary vector field and A ? the tensor of rank 2 obtained
by covariant differentiation. It may be written in the two forms
l\^ r - (96)
Similarly the tensor of rank 3 obtained by a further covariant differentia-
tion can be written in the two forms
n _ ^ a k t l rr r r n - ^ ak > J r n r F r n (W\
a k,lm *fafi~~~ L km a r,l~~ l lrn a k,r ~ ~^rn~~ r ' km '~~ 1 lm a k,r* ( Jl )
where a r ;Z is obtained from (78) or (87).
Using the first expression (96) in the first form of a klm in (97), we
get an expression for a k tlm which is a linear function of a l and its first
and second derivatives. However, if we subtract the tensor a k tml obtained
by interchanging the order of the covariant differentiations, the deriva-
tives of a disappear and we get simply
where the coefficients of a t are given by
pi _ f_fc! km pi pr pt pr /QQ\
Wm ~~ ftr"* -- ^f~ m kl ~ rl km '
Using instead the second expressions (96) and (97), we get in the same
"^,*aS ' (100)
IX, 107 PERMANENT GRAVITATIONAL FIELDS 285
where
jf ___ f_2_f.W v*i,km [ pr p pr p
Since the left-hand sides of (98) and (100) are the components of a
tensor for any vector field a,, the quantities R'u m and K lM , n must be the
components of the same tensor of rank 4, i.e.
*'u = ff'-^H. (102)
The tensor K lUm is called the Riemann-Cfuistoffel curvature tensor. The
equations (98) represent the commutation law for the covariant differ-
entiation of a vector field. The corresponding law for the covariant
differentiation of a tensor field t lk is easily seen to be
The geometrical meaning of the curvature tensor becomes apparent if
one considers a parallel displacement of a vector at along the contour
of an infinitesimal parallelogram defined by two infinitesimal vectors
(dx l ), (8x l ). As mentioned in 105, the vector a* 1 resulting from this
process will in general differ from the vector a 1 . By means of the parallel
displacement laws (62) and (67) it may now be verified by elementary
calculations that the differences
Aa l a* 1 a 1 , Aa ~ a* t a i
between the components of these vectors are given by
Aa< = \K\ lm <* do*", Aa t = i iWlIl a* d<fi", (104)
where do 2m = dtfx m dx m y*.
In a flat space, where we can introduce a system of coordinates in
which the components of the metric tensor are constant, we have
bviOU8ly R Mm = 0- (105)
This equation is thus a necessary condition for the space to be a flat
space. It is, however, also a sufficient condition; for if (105) holds at all
points, it is possible to find a transformation (9) which makes the
transformed components g' lfc independent of the space-time variables
(x /l ) (see Appendix 6).
286 PERMANENT GRAVITATIONAL FIELDS IX, 107
From (101) it follows immediately that the curvature tensor satisfies
the relations
k +-R lmkl = Q. (1066)
Besides these algebraic relations the curvature tensor satisfies a 3^t of
differential identities which may be obtained in the following way.|
On account of the 'general rule for covariant differentiation of a con-
tracted product, we get from (98) by covariant differentiation
a k,lmn a k',mln == ~~ ^klnnn a i~~ ^ klm a i t )i-
Adding to this equation the two equations obtained by cyclic permuta-
tion of the indices /, m, n, we get
Each of the three brackets on the left-hand side may be transformed by
means of the equations (103) applied to the tensors a ktl , a k . ttn , and a kttl >
respectively. In this way we get six terms, three of which cancel on
account of (1066), while the other three cancel with the terms in the
second bracket on the right-hand side of (107). Hence we get
Since this equation must hold for an arbitrary vector field a t we are led
to the Bianchi identities
m = 0. (108)
On account of the identities (106) the number of algebraically indepen-
dent components of the curvature tensor is 20 in a four-dimensional
space, 6 in a three-dimensional space, and 1 in a two-dimensional space.
108. The contracted forms of the curvature tensor
By contraction of the tensor JR\ ?W of rank 4 we get a tensor of rank 2
which, on account of (106 a), may be written in the different forms
X* == ^rk = -^fcr = ~^rk ^ *,*% ^ ^*rr (109)
As the last expression is equal to the first with i and k interchanged, this
contracted curvature tensor is obviously symmetrical
** = ** (no)
By further contraction we get the curvature scalar
* = *t = 0**, fc . (in)
t See, e.g., P. G, Bergmann, Introduction to the Theory of Relativity, New York, 1942,
p. 169.
IX, 108 PERMANENT GRAVITATIONAL FIELDS 287
Contracting the Bianchi identities (108) with respect to the indices
i and I, and using the relations (109) and (106 a), we get
*m, + X'km.>-Bt Ht m = Or **,- &",- X*,* = 0.
Further contraction with respect to the indices k and m gives
R n -2Bt - o
or, by multiplication by g ul and application of (86) and of the general
rule for covariant differentiation of products,
(#*- iy'^U = o. (112)
This equation expresses the fact that the covariant divergence of the
symmetrical tensor R k \g* k R (113)
is zero. On account of the symmetry property, this tensor has ten inde-
pendent components only.
From (109), (99), and (69) we get the following explicit expressions
for R lk -
_ ! fc , _
~^ ~" + llkr lk ' ( '
X
THE INFLUENCE OF GRAVITATIONAL FIELDS
ON PHYSICAL PHENOMENA
109. Mechanics of free particles in the presence of gravitational
fields
BY means of the formalism of the general tensor calculus and the
assumption made at the end of 104, based on the principle of equiva-
lence, the physical laws of the special theory of relativity can now be
generalized in an unambiguous way. Since the tensor equations of the
special theory are assumed to hold in a local system of inertia, i.e. in a
geodesic system of space-time coordinates, the problem of finding, for
instance, the fundamental equations of mechanics and electrodynamics
in the presence of gravitational fields reduces to a purely geometrical
problem in 4-space.
Let us first consider the motion oi a particle in an arbitrary system of
coordinates (x l ). Let , ,/ \ /i\
v ' x l = x*(r) (1)
be the equation of the time track of the motion, r being the proper time
of the particle measured by a standard clock following the particle. The
contravariant components of the four- velocity are then, on account of
(VIII. 98), di
tf=* (IV.er), (2)
dr
where U L = dx^jdt are the contravariant components of the spatial
velocity and
l f
is the analogue of the Lorentz factor in the presence of a gravitational
field with the dynamical potentials (y t , x)- In a local system of inertia,
the expressions (2) become equivalent to (IV. 39).
The covariant components of the four- velocity vector are by (IX. 16)
and (VIII. 63, 64, 94)
i.e.
(4)
X, '109 ON PHYSICAL PHENOMENA 289
where u t = y lK U K are the covariant components of the three-dimensional
velocity vector, calculated from the contravariant components by
means of the spatial metric tensor y lK . By purely spatial transformations
x' L = x'i(x*) y x'*> = x* (5)
the spatial parts U l and U L of the four-velocity transform like the
contravariant and covariant components, respectively, of a vector; but
unless the system of coordinates (x l ) is time-orthogonal, U L and U L
will represent different space vectors. According to (IV. 41) we have of
course ^ [/t = _^ (6)
which also follows from the explicit expressions (2) and (4) for U l and U t .
The components of the four-acceleration in curvilinear coordinates
are obviously
DU* <W* A - DU * dU * T E7*t7' (7)
A = r- = ~j -- h I M U u > A % ~y --- l /lA . u u (t)
ar ar ar dr
obtained by covariant differentiation (IX. 80, 80") of U 1 and U i with
respect to the proper time. In a local system of inertia, (7) reduces to
(IV. 42). According to (IV. 41') we have
U t A l = U^A, = 0. (7')
The contravariant and covariant components of the four-momentum
vector are now, by (IV. 50),
P> - rfe C/% P t - 7& U 19 (8)
where $ is the proper mass of the particle, i.e. the mass measured in a
local rest system of inertia. $ is equal to the rest mass which the particle
would have when placed in a usual system of inertia.
For a free particle, i.e. a particle which is acted upon by gravitational
forces only, we have ih a local system of inertia (x l )
T-* T = '>-
dr dr
In a general system (x l ) these equations take the forms
DP
where
(10a)
- ,, tfc (106)
dr dr
represent the covariant derivatives of the four-momentum vector.
3596.60
200 INFLUENCE OF GRAVITATIONAL FIELDS X, 109
According to (IX. 80') the quantities DP 1 are equal to the differences of
the components of the momentum four-vector P i (r-\-dr) connected with
the point x l (r-\-dr) and the vector P* 1 obtained by parallel displacement
through the distance dx l = U l dr of the vector P l (r) connected with the
point x l (r). Thus the equations (9) express the fact that the four-
momentum vector at the time t-\-dr is obtained from the vector at the
time T by a parallel displacement. On account of (IX. 51), the last
equations (9) may also be written
110. Momentum and mass of a particle. Gravitational force
The equations (9) or (11) determine the motion of a material particle
in a given external gravitational field. Strictly speaking, the particle
itself will create a gravitational field which should also be described by
the functions g lk . In the present sections we assume, however, that this
field is weak in comparison with the external field so that its influence
on g lk may be neglected. The g lk may then be regarded as known
functions of the space-time coordinates (x l ).
In order to provide a better understanding of the physical meaning of
the quantities occurring in (11) we try to write these equations in the
form of three-dimensional vector equations. Let us define a spatial
vector p lt p l by the equations
p l = mu l
= mu = mu K
If p t = p t (t) is regarded as a function of the time variable t we can
construct a new space vector
the spatial co variant derivative of the space vector p L with respect to t .
Here the y^ lK are the three-dimensional Christoffel symbols formed by
means of the spatial metric tensor y lK , and the vector d c pjdt is thus the
three-dimensional analogue of the four-vector (106), (11).
Remembering that
-~ = r, y LK = g iK +r,r^ (14)
X, 110 ON PHYSICAL PHENOMENA 291
it is easily seen that the equations (11) for i = 1, 2, 3 may be written
in the form ,
if' = * = mG " (15)
where G L is a space vector depending on the dynamical gravitational
potentials (y t , x) an( i their first derivatives. (See also Appendix 7.)
If p l = mu l is interpreted as the momentum of the particle, K L must
be interpreted as the (covariant) gravitational force acting on the
particle. The proportionality factor m defined by (12) thus appears as
the inertial mass of the particle moving with the velocity u l in the
gravitational field with the potentials (y t , x). Since K L = mG L , m also
represents the gravitational mass of the particle. For a particle at rest,
the mass reduces to o
which thus represents the rest mass of a particle in a gravitational field.
For small particle velocities, where we can neglect terms of the order
ufc t the quantity O L is equal to a t given by (VIII. 95), but in general
the gravitational force K L will be a complicated expression containing
the potentials, their first derivatives, the velocity of the particle, and
even its acceleration. There is one important case, however, where the
expression for K L becomes extremely simple, viz. if the system of co-
ordinates is time-orthogonal so that the vector potentials y t = 0. In
this case it is immediately seen that the first three equations (11) are
identical with (15) if we put
<? l= _JJ* K = mG= -mgradx, (17)
OX 1
i.e. the gravitational force is connected with the scalar gravitational
potential in the same way as in Newton's theory. This holds for arbi-
trarily strong fields and for all velocities (see also Appendix 7). Further,
if y t o, the expression (12) for the 'relativistic' mass reduces to
On the other hand, for weak fields where the dynamical potentials
(y t , x) may be treated as small, we find, neglecting terms of second order
in these quantities as well as in u/c, that (11) is identical with (15) if
we put * ~
292
INFLUENCE OF GRAVITATIONAL FIELDS
X, 110
where ^ = i"""^
is the space tensor defined by (VIII. 110) for the case of weak fields.
The last term in (18) is of the type of a Coriolis force. On the rotating
disk considered in 90, we have in the system of coordinates
x i = (x, y, 2, ct)
corresponding to the form (VIII. 83) of the line element, and in the
approximation corresponding to (18)
-(
c
(18')
Hence
02-0
c
2
c
000
and for the gravitational force we get from (18)
A r t = (m^x+Zmuu 2 , ma^yZmwu 1 , 0). (18")
For small distances r from the centre and for ' non-relativistic '
particle velocities where terms of the order u 2 /c 2 can be neglected, the
gravitational force thus reduces to the usual combination of the
centrifugal force and the Coriolis force,
Usually, the y t (and their space derivatives) are smaller than or of
the same order of magnitude as x/c 2 (and its derivatives). For small
values of ujc t the last term in (18) will then be small compared with
the first term. Further, if the field is stationary or quasi- stationary,
we may also neglect the second term in (18), and the gravitational force
is again given by the Newtonian expression (17). Thus we see that, in
the case of weak fields and small velocities, the contribution of the
dynamical vector potential to the gravitational force is generally less
important than that of the scalar potential. This is due to the large value
of the constant c. The vector potential has an appreciable influence on
the motion of a particle only for strongly fluctuating gravitational fields.
If also the geometrical influence of the gravitational field is weak, so
that the physical space is approximately Euclidean, the motion of the
X, 110 ON PHYSICAL PHENOMENA 293
particle is the same as that of a particle acted upon by a force of the type
(17) in a system of inertia.
The spatial co variant derivative of the contravariant components p 1
with respect to t is defined by
Since the spatial metric tensor y lK may depend on /, d c p l jdt and d c pjdt
will not in general be the components of the same space vector. A calcu-
lation similar to that leading to (IX. 80'") gives here
or the reciprocal relation
*'*>' = yiA^/^v = & *Y** pA (19 >)
dt Y dt Y dt * Y ft * v '
where K l = y l K K
are the contravariant components of the gravitational force. The time
derivative of the norm of the momentum vector is, by (19) and (19'),
dp, , . dp 1 d c p, , . d r p L nT r , 8y ll
V+^dt = dT p -
(20)
In a system where the dynamical potentials are zero, y t = % 0,
the gravitational force vanishes, and the motion of the particle is given
by d
df* = ' (20/)
i.e. the covariant components of the momentum vectors at different
times are obtained by parallel displacements in the three-dimensional
sense. In general this does not mean, however, that the magnitude of
the momentum vector is constant in tim, for from (20) we get
Thus, p is constant only if our system of reference is rigid, i.e. if y IK
is time-independent. If the dynamical potentials are zero, we have now
m - ^ v 2 - mW - *S^ f -
~ ' ~ ~ '
294 INFLUENCE OF GRAVITATIONAL FIELDS X, 110
and, if further the frame of reference is rigid, both u and ra are constant
and the equations of motion (20') may be written
^ =, 0. (20"')
Hence, in this case the orbit of the particle is a geodesic line in physical
space, i .e. the particle is moving with constant velocity in the 'straightest*
line compatible with the geometry of the space. The motion of the
particle is thus completely analogous to the motion of a free particle bound
to move on a smooth curved two-dimensional surface in a system of
inertia where the only forces on the particle are the normal reactions
of the surface. The only essential difference is that here we have to deal
with the motfon of a particle m a curved three-dimensional space.
If the spatial metric tensor varies with /, the motion of the particle
in the gravitational field is analogous to the motion of a particle on a
smooth variable surface in a system of inertia. Thus, if the dynamical
potentials are zero, the action of the gravitational field has the character
of a 'normal reaction' from the curved three-dimensional space.
111. Total energy of a particle in a stationary gravitational field
While the first three equations of the set (11) represent the equations
of motion of the particle, the fourth equation corresponding to i = 4
must be the law of conservation of energy. In this section, we consider
only the case of stationary fields, where we have no flux of gravitational
energy, and leave the general case to a later section ( 1G). Hence the
g lk are time-independent and the equation in question becomes
^ = 0, (21)
i e. I\ is a constant of the motion in this case. The constant H cP^
which on account of (4) is of the form
may be interpreted as the total energy of the particle in the gravitational
field (see Appendix 7). For small velocities we get, neglecting terms of
second and higher order in u/c,
(23)
X, 111 ON PHYSICAL PHENOMENA 295
since the terms of first order in the velocities cancel. // represents the
rest energy of a particle in the field . If we retain the terms of second order
in u/c, we get, using (16),
H - # +|mX, (24)
and for weak fields this reduces to the usual expression for the energy
of a particle in a field with the gravitational potential x> i.e.
ff = AoC 2 +jA tt 2 +AoX- (25)
The last two terms represent the kinetic energy and the potential energy,
respectively. For large velocities such a decomposition of the energy
into a kinetic part and a potential part is not possible.
112. General point mechanics
If the particle is acted upon by non-gravitational forces also, for
instance electromagnetic forces, we have, on account of (IV. 55, 57), m
a general system of coordinates
^^=0, (27)
where F l is a four-vector which in a local system of inertia is identical
with Minkowski's four-force (IV. 54). In the first instance, the physical
meaning of the quantities F l may thus be obtained by a transformation
of the physically well-defined Minkowski four-force. The components
F l may, however, also be given a simple physical interpretation in the
system of coordinates (x l ) itself. If we put F L = r$5n we get from the
equations (27) and (2)
i.e. J\=rg, -(3.u)). - (28)
Since dt = dr.T, we find, in the same way as in 110, that the first
three equations (26) may be written
j
(29)
which shows that the space vector g with the covariant components
3f t = FJT must be interpreted as the non-gravitational force on the
particle.
296 INFLUENCE OF GRAVITATIONAL FIELDS X, 112
Further, in the case of a stationary gravitational field, the fourth equa-
tion (26) can be written T P
tf * p
dr ~ **>
or, by means of (14), (22), and (28),
^jf = (.u)=8fe* 1 . (30)
Since the right-hand side is equal to the work done by the force g per
unit time, this equation expresses the law of conservation of energy.
113. Time -orthogonal systems of coordinates. Elimination of
the dynamical potentials
As shown in 94, it is not in general possible by a transformation of the
type (VIII. 59) (i.e. without a change of the system of reference), to
ensure that the transformed vector potentials y\ or g^ vanish. The con-
dition for this to be possible is that
"oc = 0,
where W LK is the space tensor defined by (VIII. 110).
However, if we allow arbitrary changes of the system of reference,
i.e. by means of a general space- time transformation
*'* =/*(**), (31)
we can always obtain g'^ (32)
even for the most general type of gravitational fields described by
arbitrary functions g lk or g lk in the system (x 1 ).
From (IX. 7) we get, if (32) is satisfied,
n 'il n ' _ n'i*n _ & l
9 941 (J #44 d 4>
i.e. 0' t4 -0, <7' 44 --!- (33)
(/44
The transformation law of a tensor
^iH (34)
J v '
in connexion with the condition </' t4 thus leads to the equations
( ' = 1 ' 2 ' 3) - (35)
For f*(x k ) we can now choose an arbitrary function satisfying the
necessary condition (VIII. 50):
0' = J- = |, 4 fl<7"" - grad,/* grad'/ 4 < 0. (36)
X, 113 ON PHYSICAL PHENOMENA 297
For the functions f l (x) we then get the three linear partial differential
equations of first order
grad'/g = 0. (37)
Each of the functions/ 1 thus must satisfy a partial differential equation
of the type ~ f
A % = > (38)
where the quantities A 1 = grad*/ 4
may be considered known functions.
The general solution of an equation of this type contains an arbitrary
function of three variables which may be three of the independent
variables or arbitrary combinations of them. For the functions /*,
we can thus take three arbitrary independent solutions of (38), and the
transformations (31) will then lead to a system of coordinates in which
g' 1 * == 0' i4 = o or y\ 0. In this way the vector potential has been
'transformed away' and we have obtained a time-orthogonal system of
coordinates. On account of the simplifications which arise from the
disappearance of the vector potentials, in the future we often take advan-
tage of this possibility and use time-orthogonal systems of coordinates.
In the preceding discussion, the function / 4 (x) was chosen arbitrarily
apart from the very mild restrictive condition (36). We shall now see
that it is also possible, by a suitable choice of the function / 4 (a;), to make
the scalar potential vanish. On account of (34), the condition x = 0>
i.e. 0' 44 = 1/0' 44 = 1 leads to the condition
4 (39)
for the function / 4 , i.e. the norm of the gradient of the function f*(x)
must be constant and equal to 1. Geometrically this means that the
manifold of hypersurfaces defined by / 4 (#) = constant are at a constant
distance from each other. This can obviously be obtained in an infinite
number of ways, since we can choose one space-like surface arbitrarily
and construct all the consecutive surfaces such that the distance between
two surfaces measured along the normal is constant for all points on the
surface. Now, substituting the function / 4 obtained in this way in (37),
and solving these equations, we get a set of transformation functions
f i (x) which leads to a system of coordinates in which the four dynamical
298 INFLUENCE OF GRAVITATIONAL FIELDS X, 113
gravitational potentials are zero. In a system of this type, the gravita-
tional force K occurring in (15) is zero, and the motion of a free particle
is of the type discussed on p. 293. If the initial velocity of the particle
is zero, we see at once from (20") that the particle will remain at rest in
this system. The different points of reference in the system may thus be
represented by an assembly of freely falling material particles, a circum-
stance which may be used in the practical determination of the trans-
formation functions/ 1 (x).
Although a permanent gravitational field cannot be completely
transformed away except in infinitesimal regions of space-time, it is
thus always possible to eliminate the dynamical effects of a gravitational
field over finite regions of 4-space. The effect of the gravitational field
is then purely geometrical and is completely described by the spatial
metric tensor. Although very interesting from the theoretical point of
view, the possibility of transforming away the dynamical properties of
the gravitational fields is usually of little practical importance, since the
corresponding system of reference is generally not rigid and the time-
dependence of the spatial metric tensor makes a treatment of physical
phenomena in this system very complicated. In the treatment of cosmo-
logical problems in Chapter XII we shall, however, make use of this
possibility. (See 133, 134.)
114. Mechanics of continuous systems
By the same procedure as that applied in 109-12, all physical laws
of the special theory of relativity may now easily be written in a general
co variant form. In Chapter VI it was shown that the behaviour of a
closed system in the special theory could be described by a symmetrical
energy-momentum tensor T lk satisfying the equation (VI. 1) which in
the real representation may be written in the form
div,{7**} = ii = 0. (40)
The physical meaning of the different components was explained in
62. Since the equation (40) is assumed to hold in any local system of
inertia, the general co variant form of the laws of conservation of energy
and momentum must be (see IX. 92 c)
While the conservation laws of a closed system in a system of inertia
X, 114 ON PHYSICAL PHENOMENA 299
are expressed by an equation containing a sum of partial derivatives,
we see that this is not so in the general case on account of the term on
the right-hand side of (41). This indicates that the system is no longer
closed when placed in a gravitational field which itself may contribute
to the total energy and momentum. We shall return to this question
in 126.
The components of the tensor T* in an arbitrary system of coordinates
S may be obtained from the tensor T* in a local system of inertia S by
means of the law of transformation of tensors and, since the physical mean-
ing of 5T is the same as in the special theory of relativity, we get, in this
way, a physical interpretation of the T% also. In S we can apply all the
considerations of 63, 64, i.e. if the physical system is so small that lk
may be regarded as constant over the whole, region occupied by the
system, we can unambiguously define the proper centre of mass of the
p
system. Relative to S this point is moving with constant velocity and its
motion relative to S will thus be like the motion of a freely falling
particle f
The three first equations (41) represent the law of conservation of
moment urn and the fourth equation is the law of conservation of energy.
If the gravitational field is stationary, the right-hand side of the equation
(41) with i = 4 is zero. Hence
0. (42)
Appendix 8 shows that the determinant y ~ |y t J , formed by the com-
ponents of the spatial metric tensor (VIII. 64), is connected with the
determinant g = \g lh \ by the equation g = g^.y and, since g < 0,
y > 0, we have
(43)
The equation (42) may thus be written
Now, putting
Ti (45)
t A. D. Fokker, Proc. Amsterdam, 23, No. 5, 729 (1921). C. Moller, Annales de Vlnstitut
Henri Pomcare, t. XI, fasc. V, 251 (1950).
300 , INFLUENCE OF GRAVITATIONAL FIELDS X, 114
and remembering that Vy is time-independent in the case considered,
(44) may be written -7
divS + ^ = 0, (46)
where
is the three-dimensional divergence of the space vector S (cf. the analo-
gous expression (IX. 90) for the four-dimensional divergence).
The equation (46) expresses the law of conservation of energy if h and S
are interpreted as the energy density and energy flux, respectively. In a
local system of inertia, (45) is identical with the equations (VI. 2, 3).
As we shall see in 126, the momentum density is similarly given by
(47)
If the physical system is a perfect fluid, the energy-momentum tensor
has the form
(48)
where U l =^ U l (x k ) is the four-velocity of the matter at the event point
(x k ), while j5, and f) represent the invariant mass density and pressure,
respectively, measured in a local rest system of inertia. The expression
(48) transforms like a mixed tensor and, since it reduces to the expression
(VI. 104) in a local system of inertia, it must be the correct expression for
yf in every system of coordinates.
For incoherent matter we have simply
TJ" - A(f V*. (49)
By means of the general rule exemplified in (IX. 84, 85), the conservation
laws (41) can be written
(A ^U-^+(A D ^)-^./i = o- (so)
From (6) we get by covariant differentiation
and since U t U\ k = U\ U ltk ,
we have l7.J7 lffc = 0. (51)
Thus, if we multiply (50) by U l we obtain
= o. (52)
This equation, which is the generalization of the equation (IV. 211),
expresses the conservation of proper mass.
X, 114 ON PHYSICAL PHENOMENA 301
On account of (IX. 79, 80") we now have
and (50) thus reduces to
/a.^ = 0, (53)
dr
which is the generalization of (IV. 215) for the case in which the gravita-
tional forces are the only forces acting on the matter. A comparison
with (IX. 80 1V ) shows that each infinitesimal part of incoherent con-
tinuously distributed matter is moving like a freely falling particle.
From (45), (49), (2), and (4) we get for the energy density
* = -
(54)
where F is given by (3). Now consider a small piece of matter with the
volume dV Q d l d& 2 d8? in a local rest system of inertia $. On account
of the invariance of the four-dimensional volume element (IX. 47) we
o
have, since the proper time r is identical with the time in $,
S/(i</l) dx l dx*dx*dt = c d l d& 2 d$?dr.
By means of (43) and (14) we thus get for the volume dV of the material
particle
7T/ / , u .j < dV 9 dr dV
ft i/ \Jt\t //'>**/7 x**v/ / y*"
Ct/ r irT/ Cl/t*/ Cc/vt/ Ct/C/ ~
J*
< * r "IU i fT2J 7~f ~7zl
dF =
(55)
This is the generalization of the Lorentz contraction formulae in the
presence of gravitational fields. For U K = 0, we get
dV Q - dV, (56)
i.e. the volume of a small piece of matter at rest in 8 is equal to its volume
o
measured in in accordance with tha general assumption made on
p. 223.
o
If T& O = ^5<> dV Q denotes the proper mass of the particle, its total energy
H = h dV is obtained from (54) and (55)
(57)
in accordance with (22).
302 INFLUENCE OF GRAVITATIONAL FIELDS X, 115
115. The electromagnetic field equations
For simplicity we confine ourselves to the case of electrodynamics in
the vacuum, the generalization of the macroscopic theory in ponderable
matter running along the same lines. By means of the generally covariant
expressions for the divergence and curl of an antisymmetrical tensor
(IX. 93, 94), we can write Maxwell's equations (V. 9, 13, 16) in an arbitrary
system of coordinates S in the form
0-^1* , $ F ki , 3F h o / * Q x
+ + = ' ( }
(59)
Here, F lk F kl is the electromagnetic field tensor, s l is the four-
current density, p is the charge density measured in a local rest system
of inertia, and f/* is the four- velocity of the electric charge. In every
local system of inertia S, the equations (58) and (59) reduce to the
equations (13), (16), or (1) in Chapter V.
On account of the antisymmetry of F lk we get at once from (586)
} s ' (60)
which is the general form of the continuity equation of electric charge.
Now consider a system which at the time t x*/c lies entirely inside a
finite region of volume V in 3-space. Multiplying (60) by *J\g\ and
integrating over the space coordinates x l , x 2 , x* we get, since the first
three terms in the summation over i are partial differential coefficients
with respect to x l , x 2 , rr 3 ,
v
Thus the quantity
~ f Vlff | s* dx*dx*dx* - 0. (61)
= [ h + ^s*J 7 dx l dx*dx*== [(
dV (62)
is a constant in time, which must be interpreted as the total electric
charge of the system. Hence
(63)
X, 115 ON PHYSICAL PHENOMENA 303
must be the charge density in the system $. Now consider the charge
o
p dV connected with a volume element dV\ if dV Q represents the corre-
sponding volume in the local rest system, we get from (63) and (55)
p dV = p dV Q , (64)
i.e. the electric charge of the small piece of matter considered is indepen-
dent of the system of coordinates.
The invariance of the total electric charge also follows directly from the
continuity equation (60). Consider two arbitrary systems of coordinates
S and S' and the integrals
e= J J\g\ s* dx l dx*dz* (65)
x 4 -a
e' = I <J\g'\8'*dx' l dx'*dx'* (65')
over the two regions defined by
# 4 = constant = a and #' 4 = constant 6,
respectively. On account of the time-independence of the integrals (65)
and (65') we can, without any change in the values of e and e', choose b
so that the regions #' 4 = 6 and x* a do not overlap inside the tube
in 4-space in which the charge density is different from zero. Then we
can introduce a third system of coordinates /8" which coincides with S
inside the region # 4 a and with 8' inside the region #' 4 6. As the
equation (61) holds also in the system 8", we have
J J\g* | s" 4 dx" l dx"*dx"* - f Vl/ 1 <*" 4 dx" l dx" z dx"* (66)
x'*^x*^a x'*^jr' 4 = b
and, since x" 1 = x 1 on the hypersurface x 4 = a and x' fi = x' 1 in the region
#' 4 = 6, (66) leads to the equation
e - e ; , , (67)
which expresses the invariance of the total charge.
On account of (59), (63), and (2) we have
H,
(68)
- -=0 or divpu + - = 0, (69)
Vy dt H ^Vy dt
and the continuity equation (60) can be written
304 INFLUENCE OF GRAVITATIONAL FIELDS X, 115
which is the three-dimensional form of the continuity equation when the
spatial metric is time-dependent. For rigid frames of reference, where y
does not depend on t, (69) reduces to the usual form
In order to obtain a closer understanding of the physical meaning of
the tensor components F lk we shall now write the Maxwell equations
(58) m the form of three-dimensional vector equations. For simplicity
we assume that our system of coordinates is time-orthogonal, i.e.
which, as was shown in 113, can always be obtained by a suitable choice
of the space-time coordinates.
Let us now introduce two antisymmetrical space tensors H IK and
B IK and two space vectors D L and E c by the equations
Fut _ _#"_ , .. __ '__ _ __ &_
"V(l+2x/c 2 ) ' " V(l+2x/ a ) ' 1+2X/C 8 '
(71)
These quantities are thus connected by the equations
H IK R
BIK ___ __ _ ___ T) __ __ ^79^
^V(l + 2 X /c 2 )' Vtl+Sx/c*)' V ^
From (IX. 27) and the corresponding equations for spatial tensors we
then get, using (70) and (71),
(1 ( j vim (J .,, php __ v
IK - - i/tjy/cw* 1/tXy KH L ri
With these notations, the equations (58 a), (586) may be written m
the form of three-dimensional tensor equations:
U - o,
r. fir. i
(74)
vy c bt
X, 115 ON PHYSICAL PHENOMENA 305
where o D n $ D
curl - IK * A At
curiu
_
, ^ # &EI
curl...E = - * -- -
ftr 1 iWAC
,
(75)
divD =
are the three-dimensional covariant differential operators. Further,
defining the axial vectors dual to the antisymmetrical tensors LK ,
H LK by
**- -^- *-^| ,76,
//! = Vy// 23 , // 2 = Vy// 31 , // 3 = Vy^ 12 J
(see Appendix 5), Maxwell's equations (74) take the familiar form
div B ~~
CV/y * }, (77)
, T 1 (VyD) pll ,. ~.
curl H r - = ! div D = p
\y C ct C
where curl E and curlH are the vectors dual to the tensors curl tK E and
curl l/c H, respectively. For a rigid system of reference, where y is time-
independent, these vector equations are of the same form as Maxwell's
phenomenological equations in ponderable matter and, since (72) can
be written D = E, B - M H, (78a)
:, (786)
we see that the gravitational field, besides its influence on the spatial
geometry, acts like a medium with the dielectric and magnetic con-
stants (78*6).
116. Electromagnetic force and energy -momentum tensor
The electromagnetic four-force acting on a particle with charge e
is, according to (V. 85),
F, = * c F*W<. (79)
For the components of this vector we get, by means of (2) and (73),
3595.80
306 INFLUENCE OF GRAVITATIONAL FIELDS X, 116
A comparison with (28) shows that the electromagnetic force on a charged
moving particle is
(80)
which, by means of the axial vector B defined by (76), may be written
(81)
where u X B is the vector product with the covariant components
(u x B) t - Vy(tt 2 3 -w 3 ,B 2 , u*B l -u l *> u l B 2 -u*JB l ) (81')
(see Appendix 5).
For a system with continuously distributed charge the force on the
charge in a volume element dV is given by (81 ) with e = p dV. Therefore
the force density f must be
(82)
which is formally identical with the expression for the ponderomotive
force (VII. 73) following from Minkowski's energy-momentum tensor in
ponderable matter.
For the four-force density we get from (V. 93)
/> = V, (83)
which, on account of (68) and (73), leads to the expressions
*><*.") , (84)
- lf(E l +B
~ \ V(T+
which represents the generalization of (IV. 214) in the presence of a
gravitational field.
According to (V. 96) the equations of motion, in the general system S,
are
If we multiply this equation by the proper volume dV of a small piece of
o
charged matter, then, since the proper mass /x dV is conserved by this
type of force, we get, using (55),
(87)
X, 116 ON PHYSICAL PHENOMENA 307
where P l /l dV Q U t is the four-momentum of the small particle. On
account of (85) and (14), these equations may also be written
%? = K+f dV, *Z = (f . u) dV, (88)
at at
similar to (29) and (30).
The first equations are the equations of motion of the small piece of
matter under the influence of the gravitational force K and the electro-
magnetic force f dV, the last equation is the law of conservation of
energy.
From the validity of the equations (V. 105, 106) in a local system of
inertia we get m the general system S
fl = -di
where 5* = Sf = S\ = F F* l -^\(F lm F 1 ) (90)
is the electromagnetic energy-momentum tensor. The equation (89) is a
consequence of the field equations (58) together with (83),
Using (71), (73), and (76) in (90) we get for the different components
of the tensor S*
(91 a)
where E x H and D X B are the vector products defined by (8 1') or by
the equations (5) and (6) in Appendix 5.
From (45), (47), and (91) we get for the electromagnetic energy flux,
-energy density, and momentum density
s = c(ExH), h = KE.D+H.B), (92)
g = -(DxB). (93)
c
These expressions are in complete agreement with the formulae (VII.
70, 71, 72) following from Minkowski's energy-momentum tensor
(VII. 68) in ponderable matter, as distinct from Abraham's tensor which
gives g = (ExH)/c instead of (93).
308 INFLUENCE OF GRAVITATIONAL FIELDS X, 117
1 17. Propagation of light in a static gravitational field. Fermat's
principle
In a static gravitational field, where y t = and y lK and x are indepen-
dent of the time variable , the electromagnetic field equations (77) take
the form of Maxwell's phenornenological equations in a medium at rest
with i
Since the spatial geometry may be regarded as Euclidean in a sufficiently
small part of space, it follows from these equations that a light wave with
sufficiently small wave-length, i.e. in the limit of geometrical optics, will
propagate with the velocity
w = - = -p-- = c /(l+-%\ (94)
in accordance with the equation (VIII. 70). The trajectory of a light ray
in a static gravitational field is therefore determined m the same way as
in an inhomogeneous refractive body, i,e. by Fermat's principle (see
10), according to which the time which the light takes to travel between
two points A and B in space is a miiumum for the actual path chosen
by the light ray. Mathematically this is expressed by the variational
principle 1} ^
8 J = 8j *^- r) ==0 (95)
for all variations of the curve connecting A and B. Since y lK = g iK in
our case, (95) may be written
8 L(x i 9 x i )dX --- (96)
with t
Here, A is an arbitrary parameter in a parametric representation x l = x l (X)
of the curve considered. According to the equations (21), (24) in 86, the
condition (96) is equivalent to the Euler equations
d I _ M'LJl = M
dX^ig^x*^)! 2
.A (98 v
w* dx c l '
Since the parameter A was completely arbitrary, we may choose it so
that ,
w VtoicA*^) = ~j( w = constant = 1 (99)
X, 117 ON PHYSICAL PHENOMENA 309
along the curve determined by (98). These equations then reduce to
d c u L _d ^v 1 ^^.- __L^f - __1^X MOO}
7T " dX ((/LK } 2 dx' ~~ uPSx* ~~ w*dx^ ( }
which together with (90) comj)letely determine the trajectory of the
light ray. The term 071 the right-hand side ( 1 /w 4 )grad x determines
the deviation of the light ray from the 'straightest' line. According to
(17) this term is equal to (l//r 4 )G, where G is the gravitational force
on a unit mass, thus the light ray \vill he a straightest line only if the
gravitational force K is zero.
The same result may he obtained from the equations (VIII. 87) for
the time track of a light ray/|* Since y t 0, w ~= c^( j/ 44 ), the last of
these equations may be* written
field d
= 0. (101)
Further, we get from the first equations with i 4 in the case of a static
!^\-o
; a dX) ~~ '
which shows that \r-.i - constant
along the time track. By a proper choice of A this constant can be made
equal to 1. Hence, w *.t-~ 1, (102)
and, on account of (101),
which shows that the parameter A defined by (102) is identical with the
parameter defined by (99). For i = 1, 2, 3 we then get from (VIII. 87)
-~ LK =^ ~- ----- ,
d\ iK 2 3x L 2 fa 1
which, on account of (102), is identical with (100). Format's principle
of least time thus holds in every static gravitational field.
t T. Lovi-Civita, fiend. Acca<l Lmcei (.5), No 26 (1017), p. 458; Nuorv Ctmenlo (6),
16, 105 (1918), H. Weyl, Ann. d Phys 54, 117 (1917).
XI
THE FUNDAMENTAL LAWS OF GRAVITATION
IN THE GENERAL THEORY OF RELATIVITY
118. The gravitational field equations
IN the preceding sections we considered the influence of a given gravita-
tional field on physical phenomena. We now turn to the most important
problem in gravitational theory, which consists in finding the general
equations determining the gravitational field variables (y (K , y t , x) r the
g lk from a given distribution of mass. After several attempts, this
problem was finally solved by Einsteinf in 1915. In Newton's theory of
gravitation, the corresponding problem may be stated in the form of
Poisson's equation A = 4 & (1)
where k = 6-664 x 10~ 8 cm. 3 gm.- 1 sec.~ 2 (2)
is the gravitational constant. This equation enables us to calculate the
gravitational potential x when the mass density /i is given as a function
of the space coordinates.
On account of the equivalence of mass and energy, we must assume
that any energy distribution, thus for instance an electromagnetic
field, will create a gravitational field. Now, the energy density of any
physical system is given by the component T 44 of the energy-momentum
tensor T lk of the system, while x = 2 c2 (l9w) 1S connected with the
component 44 of the metric tensor; thus, equation (1) expresses the fact
that a certain differential operator of second order, acting on </ 44 , is
proportional to T 44 . Since the general field equations must be covariant,
and since different components ofT lk are mixed up by a transformation of
coordinates, it is natural to assume that the general field equations are of
the form ^ _ T ,*
m ik K -*iki ('*)
where K is a universal constant, and M lk is a tensor of rank 2 depending
on the metric tensor g lk and its first and second derivatives only. Since
the equation (3) for weak fields must reduce to Poisson's equation (1),
M lk must be linear in the second derivatives of g lk , and the only possible
expression for M lk is then of the form
M ik z= R lk +c l R.g lk +c^g lk , (4)
where c t and c 2 are constants, while R lk and R are the contracted forms
of the Riemann-Christoffel curvature tensor defined by (IX. Ill, 114).
t A. Einstein, Berl. Bcr. % pp. 778, 799, 844 (1915), Ann. d. Fhys. 49, 769 (1916).
XI, 118 THE GENERAL THEORY OF RELATIVITY 311
On account of the symmetry properties of the tensors occurring in
(4) the equations (3) represent ten differential equations for the ten func-
tions g lk . A simple consideration shows, however, that these ten equations
cannot be independent (Hilbert).*)* Consider, for instance, the special
case of empty space, where T lk 0. The equations (3) then reduce to
M lk = 0. (5)
If these equations were independent, the ten equations (5) would in a
definite system of coordinates (x l ) allow us to determine uniquely the
functions g lk (xt) throughout the whole 4-space, when the values of g lk
and f dg lk \'dy} are given on a hypersurface
^ = constant = a. (6)
If w r e now introduce a new system of coordinates x' 1 by
x' 1 = x'*(x), x l = x*(x')> (7)
the transformed functions
satisfy a set of differential equations
M' lk - 0, (9)
where, on account of the covariance of the equations (5), M' lk are the
same functions of g' lk , dg' lk /dx' 1 , d*g' lk /dx' l dx' m as M^ k of g lk , dg lk /dz? t
d*g lk /dx l dx m . Thus, by the same argument as before, the equations
(9) enable us to determine uniquely the functions g\ k (x') from the values
of g\ k and dg r lk /dx' 1 on the hypersurface # 4 x*(x') = a, and if we choose
the transformations (7) such that x' 1 ~ x 1 in the vicinity of the surface
# 4 .= a, but arbitrary elsewhere, we have
on this surface, and g lk (x' 1 ) must consequently be the same function of
the variables (x /l ) as g^x 1 ) is of (x 1 ). This is, however, in contradiction to
(8), which shows that the dependence of g' rk on (x' 1 ) for points sufficiently
far from the surface x 4 = a will, in general, differ from the functional
relationship between g lk and (x 1 ).
Thus, if the field equations are to be co variant, we must assume that
the quantities M lk on the left-hand side of (5) and (3) satisfy four identi-
ties. This means that the solutions g lk of the field equations contain
four arbitrary functions corresponding to the four arbitrary functions in
t D. Hilbert, GMt. Nachr., p. 395 (1915).
312 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 118
the transformations (7), which only change our space-time description,
but not the physical system which produces the gravitational field.
In fact, as seen in 113, it is always possible by a proper choice of the
space-time coordinates to ensure that the four functions l4 have the
prescribed values S l4 throughout space-time. The six independent
equations which remain after the introduction of the four identities
involving the quantities 3/ ?A . are thus just sufficient to determine the six
remaining components g lK of the metric tensor.
As we have seen in 114, the theorems of conservation of energy and
momentum of a material system in a general system of coordinates have
thef rm Th = div 4 (T{) - =0. (10)
Thus, if we assume that the four identities involving the components of
the tensor M lk are ^ ^ ^ ^
the conservation theorem for a material system is a consequence of the
field equations (3) in the same way as the conservation of electric charge
(X. 60) follows from iMaxwelTs equations (X. 58).
We therefore assume that the differential operators M lk on the
left-hand side of the field equations satisfy the four identities (11).
According to (IX. 86) and (IX. 1 12) we now get from (4)
and, if this is to be identically zero, we must have
q = -J.
Thus, putting c 2 =^- - A, where A is a universal constant, we get
M lk . K, k -\R< J>k ~\g tk , (4')
and the field equations (3) take the form
M,k-- K lk -\Eg lk -^ tk -= - K T, k . (12)
Writing down this tensor equation in mixed components, we get
3/{ tf?--|#8{ -AS* = -*!*. (13)
By contraction we obtain, since R\ ~ R, S[ = 4,
j?-f 4A - KT, (14)
where T = T\ (15)
is the invariant obtained by contraction of the energy-momentum tensor
T lk . Eliminating R by means of (14) the field equations (12) can also be
written in the form
. (16)
XI, 119 THE GENERAL THEORY OF RELATIVITY 313
119. The linear approximation for weak fields
The field equations (12) and ( 1 6) are in general non-linear partial differ-
ential equations in the functions g lk . However, for weak gravitational
fields, the field equations may approximately be replaced by a set of linear
differential equations, f A weak field means that ue can introduce a
system of space-time coordinates in which the metric tensor is of the form
f/,A - ,A+A,A. (17)
where O lk is the constant metric tensor (VIII. 41) of the special theory,
and the h lk and their derivatives are small quantities whose squares
may be neglected. The ( 1 hristoffel symbols (IX. 50) will then obviously
be small of the first order and, in the curvature tensor (IX. 99, 114), we
can neglect all terms depending on the squares of the Christoffel sym-
bols. Hence we get
R _^
'* ~ ~
~dx k
= rs JL p^+^^M--" /%< , ^A_J
2 ftE^af rV dx 8 ] 2 dx r \8x k ~ t dx l dx 8
^ .a* w , i/ &*. &% a'AH
2 ( ] x r dx s f 2\dx'dx k dx r dx k dx l dx r )'
where we have put
hi = G rs h ls , h - h r r = G'h rs . (19)
Let us now first consider the case of a static distribution of matter.
This means that among the systems of space-time coordinates of the
type (17) there will be some in which the field variables li tk and the com-
ponents of the material energy -momentum tensor are independent of
t. For the component jR 44 we then get from (18)
where x is the scalar gravitational potential defined by (VIII. 94) and
A is Laplace's operator. If, in the case of matter at rest, we neglect the
small contributions of the elastic stresses to the energy-momentum tensor
we have, to a first approximation,
(21)
}
f A. Einstein, Berl. Ber., p. 688 (1916).
314 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 119
Thus we get from (16) with i k = 4
= >. (22)
Now we know that Poisson's equation ( 1 ) is a good approximation for all
static and quasi-static gravitational fields inside the solar system. We
can therefore conclude that the constant A must be so small that the
A-term may be neglected for all gravitational phenomena inside the
planetary system of the sun. The A-term can be of importance only
for cosmological problems and in all other cases we shall therefore put
A equal to zero. The field equations (12) then reduce to
R lk -\Rg, k ^-*T lk . (23)
Further, comparison of (22) and (1) shows that the constant K must
be connected with the gravitational constant k by the equation
*c 2 = = 1-87 X 10- 27 cm gm.- 1 (24)
c
Returning now to the case of a general weak field, we first remark
that the system of coordinates corresponding to the form (17) with h lk
small to the first order is still to a large extent arbitrary. Any trans-
formation of the type ^ ^ X ^8x l 9 (25)
where Sx J (#0 is a function which is small to the first order, will lead to an
expression for the transformed metric tensor of the same form as (17).
Since the left-hand side of (23) is small to the first order, the same must
be the case for KT lk , which means that, to the approximation considered,
T lk may be treated as invariant under the transformations (25). Hence
the components of T lk may be put equal to the corresponding expressions
in the local system of inertia.
The expression (18) for R lk may now be written in the form
with x? = A*-i8*A, Xi* = *i*~ !#,**, * = *! (27)
The quantities Xi k , X? transform like the components of a tensor by linear
transformations of the space-time coordinates. On the other hand, it
is always possiblef by a transformation of the type (25) to ensure that
the x* satisfy the equations
I? = - (28)
t See D. Hilbert, Gfitt. Nachr., p. 53 (1917), and the solution (31) which satisfies (28).
XI, 119 THE GENERAL THEORY OF RELATIVITY 315
For R lk and R we then get
*tfc = 4DAifc, /Z=*J = inA, (29)
and the field equations (23) reduce to
These equations are of the same form as the equations (V. 26) for the
electromagnetic potentials, and the solutions of (30) which vanish at
infinity are therefore analogous to the ' retarded ' potentials (V. 46)
X,i(*S - l~ J ~^^ C) <te'WW, (31)
where r /( T (x l x' l )A. If we can prove that the functions defined
** l i=lT2,3 '
by (3 1 ) also satisfy the conditions (28), they must be the required solutions
of the approximate field equations (23). The proof that the solutions
(31) really satisfy theequations(28)runsexactly as the proof of the validity
of the Lorentz condition for the retarded electromagnetic potentials.
In the latter case, this proof is based on the law of conservation of electric
charge (see V. 41 ). In the present case, the validity of (28) follows in the
same way from the law of conservation of energy arid momentum which,
to the approximation considered, may be written
3T k
a? = - < 32)
From (31) we can now also find explicit expressions for the, quantities
h lk . First we get from (27)
X\ = h\~2h = -h, h lk = x^k+\G ^k h = xifc-l^xj; (33)
hence, by means of (31),
T'dV
*"-/
h tk = f g?
ZTT J
and h lk = -?* - '* dP, (34)
where the prime in T' lk and T r means that these quantities have to be
taken at the place (x' L ) of the volume element dV = dx' l dx' 2 dx' 3 and at
the retarded time tr/c.
120* Simple applications of the linear equations for weak fields.
The relativity of centrifugal forces and Coriolis forces
To begin with let us consider a static distribution of matter, where the
mass density /a /*(#, y, z) is a given function of the space coordinates
316 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 120
x' = (x,y,z). In this case we have, according to (21) and (34),
^ = 8,48^, -A 2 . T=-A 2
(35)
7 *c 2 C nO(x' 9 y' 9 z')dx'dy'dz'
"44 = 4- I : [X--X'|
h KC 2 C p<>(x r 9 i/' 9 z')dx'd!/'dz'^ _, 8
and for the gravitational potential we get, on account of (24), the usual
expression of the Newtonian theory
(36)
Hence the line element is of the form
ds 2 = (G lk +h lk ) dx*dx k
(37)
where x * s given by (36).
The spatial line element is
da^ = /I--?
(38)
hence the geometry is only approximately Euclidean and the coordinates
x, y, z are not exactly Cartesian. In general it is not possible by a change
of the space coordinates to introduce Cartesian coordinates. However,
the deviations from Euclidean geometry are in most cases too small to be
measured. At the surface of the earth, for instance, the quantity 2^/c 2
is of the order of magnitude 10~ 9 .
For a system of material particles with the masses M\, M%,... situated
at the places X^ X 2 ,..., respectively, we get from (36)
(39)
For a single particle the potential and the line element are spherically
symmetrical. We have
(2k
1 + ^
r=|x-X|
(40)
XI, 120 THE GENERAL THEORY OF RELATIVITY 317
- In the same way, the case of a stationary mass current distribution
may be treated by means of Einstein's approximate equations (34).
Thirring and Lensef calculated, for instance, the influence of the rotation
of a central astronomical body on the gravitational field and the corre-
sponding effects on the motion of the satellites. All such effects are too
small to be observed J and we shall not consider them here.
There is one effect of this kind, however, which, although small, is of
theoretical importance since it throws new light on the nature and
origin of the centrifugal and Coriolis forces appearing in a rotating
system of coordinates N. According to the idea of Einstein underlying
the general principle of relativity (cf. 82), these forces are gravita-
tional forces originating from the rotation of the distant celestial
masses relative to $, and such 'non-permanent 5 gravitational fields
should satisfy the same general field equations as the permanent fields.
The approximate solutions (34) for weak fields do not directly allow us to
treat the effects of the distant celestial masses, but we may expect that
a rotating spherical shell of uniform mass density will produce effects
inside the shell similar to the rotation of the distant celestial masses.
If the shell is at rest, the potential x given by (36) is constant inside
the shell and has the value x = kM^R, i.e.
k >v *-r 2 M
* = - K *%, (41)
c 2 4?r H
where M is the total mass and R the radius of the shell.
Apart from the constants 1 2^/c 2 and 1 + 2^/c 2 , which can be re-
moved by a simple change of scale of the space and time coordinates, the
line element (37) thus has the same form inside the shell as in the special
theory of relativity. In the case of a shell moving with constant velocity,
which can be reduced to the former case by a Lorentz transformation,
the line element inside the shell must therefore again be of the special
relativity type. This is also easily seen directly from (34), using the
expression for T ik in the case of stationary rectilinear motion of the
matter. This illustrates the fact that the uniform rectilinear motion of
the distant celestial masses relative to the different systems of inertia
does not give rise to any gravitational forces in these systems.
For a rotating shell of matter, however, Thirring found the interesting
result that the field in the interior of the shell, as determined by the
equations (30), is similar to the field in a rotating system of coordinates,
t H. Thirring and J. Lense, PJnjs. ZS 19, 156 (1918).
j Do Sitter, Monthly Notices, 76, 699 (1916), 77, 155 (1916).
H. Thirring, Phy*. ZS. 19, 33 (1918); 22, 29 (1921).
318 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 120
thus leading to gravitational forces similar to the usual centrifugal and
Coriolis forces. We shall here consider the somewhat simpler case of a
rotating massive ring of rest mass M Q and radius R, which is rotating
clockwise in the xy-pl&ne with angular velocity to. If (#',?/', 0) are the
coordinates of a point on the ring, we have
( 42 >
/7T44
-*
From (IV. 198) we get for the total rest mass M of the ring
Hence, by means of (34),
A lfc (*,y, 2 ) = C f^-*, (43)
J P
where p - J{( x - x ')* + ( y - y >)*+ z *} (44)
is the distance between the point (x, y, z) and the line element ds on the
ring. Further, 2
and
2 2 c 2 c 2
y'x' o> 2 , 1
-
c
1 1 JP2 2
i-i*-?.. o
o i+i
c c 2 ' 2 c 2
(45')
We shall confine ourselves to the consideration of points (x, y, z) whose
distance from the origin is small compared with R. This means that we
can use an expansion of !//> in terms of x/R, y/R, z/R. Neglecting all
terms of higher than second order in these quantities and putting
x = COS, y =
XI, 120 THE GENERAL THEORY OF RELATIVITY 319
we get
1 1 /, 1 # 2 +v 2 +z 2 , a a/ , V <w , 3 x 2
a a/ , V <w , 3 x
s 00 ^ +r mi? +2 ^
'+- 2 (46)
Using this together with ds = R d&' in (43) the integration over
#' from to 27T can be easily performed. We shall- only write down the
expressions obtained for the components g i4 <7 l4 -f/i l4 , which determine
the dynamical potentials, i.e.
CTTOJ
^34==
(47)
where 3f == 3f (l RW/c 2 )-*
is the relativistic mass of the shell. These expressions should hold for all
values of w for which Rwjc < 1, the only approximations introduced
being those arising from the use of the equations for weak fields and the
expansion of 1/p in terms of x/R, yjR, zjR.
For a ring of mass M at rest we get in the same way
In this case, the scalar potential is thus not constant as inside a closed
shell. Hence, the contribution to the dynamical potentials arising from
the rotation of the ring is, apart from the unimportant constant term
E 2 a> 2
in the expression for gr 44 ,
MKC* a) , AN \
<-->
320 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 120
A comparison with (X. 18') shows that the gravitational force on a
moving test particle inside a heavy wheel rotating clockwise in the in/-
plane is of the same type as in a system of coordinates rotating counter-
clockwise. It is true that on account of the smallness of the quantity
MKC 2 /^7rfi the effect is too small to be measured, which explains the
negative result of the experiment actually performed by Friedlander ,f
b'ut anyhow the above considerations suggest a connexion between the
gravitational constant /c, the total mass M in the world, and the mean
distance R of the distant celestial masses, of the type
MKC* .
It is interesting that the dependence on the angular velocity of the
gravitational forces inside a rotating wheel is exactly the same as in a
rotating system of reference. The vector potentials y t in (48), which give
rise to the Coriohs forces, are even of the usual form as regards their
dependence on the coordinates (x,y,z). On the other hand, the scalar
potential in (48) contains besides the usual term x*+y 2 a term depending
on z*. This term gives rise to an axial component of the 'centrifugal'
force which tends to drag a test particle into the plane of the rotating
wheel. One could think that this unexpected deviation from the usual
centrifugal force is due to the particular mass distribution which we
have considered. However, the calculations of Thin-ing show that the
same effect appears inside a homogeneous rotating shell. The purely
radial character of the usual centrifugal force therefore rather indicates
that the approximate field equations (30), which require explicit assump-
tions regarding the boundary conditions at infinity in order to give
unique solutions, do not give an adequate description of the world as a
whole. Firstly, the exact equations (12) are non-linear and, secondly,
they contain the A-term which for cosmologicacl distances may be of
importance. Jn fact, this term entirely changes the character of the
field equations, in particular as regards the question of the boundary
conditions (see 132). But even if we confine ourselves to the linear
approximation of weak fields, a simple calculation shows that the
use of the A-term introduces into the solutions of the equations for
the case of a rotating wheel extra terms which are of the same type
as those occurring in (48), but multiplied by \R 2 . The terms depending
on (jJ 2 +j/ 2 ) and 2 2 do not, however, have the same ratio as in (48); there-
l, (50)
t B. and T. Fnedldnder, Absolute und relative Bewegung, Berlin, 1896.
XI, 120 THE GENERAL THEORY OF RELATIVITY 321
it is understandable that the terms in x which contain z 2 may cancel
and that we are left with a purely radial centrifugal force. This con-
sideration is, of course, very rough, since the linear equations probably
represent a bad approximation when applied to the world as a whole;
but, as we shall see in 133, the relations (49) and (50) also follow from
Einstein's solution of the exact gravitational equations applied to an
ideal model of the universe.
Finally, if we treat the case of a non- stationary distribution of matter
by means of (30) and (34), the close analogy to the wave equations of
electrodynamics leads at once to the result that fluctuating matter in
general gives rise to the emission of gravitational waves travelling
with the velocity of light and carrying with them a certain amount of
energy. As shown by Einstein, f the gravitational energy (see 126)
emitted in this way is, however, too small to give any measurable
astronomical effect.
121. Equivalent systems of coordinates. Systems with spherical
symmetry
Let (x l ) be an arbitrary system of coordinates with the metric tensor
g lk g lk (xl). If we introduce a new system of coordinates x' 1 by
x' 1 = x' l (x*), (51)
the transformed components of the metric tensor
f\^l wm
^ (x>r) = W* w***W < 52)
are not in general form-invariant functions of the coordinates, i.e.
g\ k is generally not the same function of the transformed variables (x' 1 )
as are the g lk of the variables (x 1 ).
Two systems of coordinates (x 1 ) and (x' 1 ) for which the components
of the metric tensor are form-invariant functions of the space-time
coordinates under the transformation (51) may be called equivalent, since
any physical process will have the same course of development in the
two systems. The existence of equivalent systems of coordinates imposes
a certain condition on the gravitational field, since the functions (7i&(#0
obviously must satisfy the functional equations
In some cases the gravitational field variables g lk are form-invariant
under a whole group of transformations. This is, for instance, the case
3595.60
t A. Emstem, Berl. Ber , p. 154 (1918).
Y
322 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 121
for all non-permanent gravitational fields, for, if we first introduce
pseudo- Cartesian coordinates X 1 by the transformations
X*=/^), (54)
we have ds* - g lk dx l dx k = G lk dX*dX k . (55)
Further, by performing a Lorentz transformation
JT* = A> k X k (56)
and afterwards introducing new space-time coordinates (x' 1 ) by means of
the transformations v// __ ft{ ,K / r7 v
.A J (% ) (*< )
with the same functions/ 1 as in (54), we have
g lk dx*dx k - G lk dX*dX k = G' lk dX'*dX' k ~ g\ k dx'*dx'*,
and, since G[ k G lk constant,
g' lk must be the same functions of the (a'*) as are the g lk of (r z ). The
transformation x' 1 x' l (x l ) defined by (54), (56), and (57), which may be
called a generalized Lorentz transformation, thus connects two equivalent
systems of coordinates, and the components y lk of the metric tensor in a
non-permanent gravitational field are form-invariant under the group
of generalized Lorentz transformations.
In the case of permanent gravitational fields, it is generally not
possible to introduce such space-time coordinates x l that the g lk are
form-invariant under the group of four-dimensional orthogonal trans-
formations, but in some important cases the gravitational potentials
are form -in variant under the sub-group of spatial orthogonal transforma-
tions. Such systems are naturally called spherically symmetric. Putting
x l (x,c) = (x,y,z,ct), the g tk are then form-invariant under any
orthogonal transformation of the three variables x, ?/, z with constant t.
Generally the coordinates (x,y,z) will not be Cartesian, the spatial geo-
metry being in general non-Euclidean, nevertheless, the line element
ds 2 = g ik dx l dx k can in this case be a function only of the well-known
foi'm-invanants of the group of three-dimensional rotations in a Eucli-
dean space, These invariants are
r = { x *+y*+ z *}\, dr \
dx 2 +dy 2 +dz 2 =. dr*+r* d0 2 +r 2 sin 2 rff 2 1. (58)
x dx-\-y dy-\-z dz = r dr, dt, and t )
Since the line element is a quadratic form in the differentials, the most
XI, 121 THE GENERAL THEORY OF RELATIVITY 323
general possible expression for ds 2 in a system with spherical symmetry
is therefore
ds 2 - F(r,t) dr 2 +G(r t t)(r 2 d0 2 +r 2 sin 2 d</> 2 ) +
+ 2H(r, t) drdt+L(r, t) dt 2 , (59)
where F, G, H, L are functions of r and t, only.
This expression may be further reduced by a suitable choice of co-
ordinates. Introducing a new variable r' instead of r by the transforma-
tion r' 2 = r*G(r 9 t), (60)
we see that the line element in the new variables is of the form
ds 2 -- M(r',t) dr f2 +r' 2 (d0 2 +sm 2 0d<f> 2 ) + 2N(r',t) dr'dlt+0(r',t) dt 2 .
(61)
The quantities o> tK defined by (VIII. 110) are obviously zero in the
present case. Hence it is possible by a simple change of rate of the co-
ordinate clocks to make the vector potentials disappear, and since
N(t',t) does not depend on and <, this may be obtained by a time
transformation of the form
t'=f(r',t), (62)
which will not affect the second term in (61). Dropping the primes, the
line element may thus be expressed in the standard form
ds 2 == a dr 2 + r 2 (d0 2 +sm 2 d<f> 2 ) bc 2 dt 2 , (63)
where a = a(r, t), b = b(r, t) = I + - (64)
C"
are functions of r and t which, on account of (VIII. 52), must be positive
for all values of r and t.
122. Static systems with spherical symmetry
If the system is static and spherically symmetrical, the functions a
and b are independent of /, and in this case the components of the tensor
M lk defined by (4') are easily calculated. With x l = (r,6,(f>,ct) we have
Vu = aM> 022 = r *> I/as = r 2 sin 2 0, 44 == -6, (65)
all other components vanishing. The coordinate system is orthogonal;
the non-vanishing components of g lk are therefore simply
(66)
324 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 12J
and the Christoffel symbols (IX. 50) are of the form
(no summation over i and fc ').
Since the g lk are independent of </> and a: 4 , we see that the only indepen-
dent non-vanishing components of T l kl T\ k are the following:
pi _ a r 1 r r 1 _ r <5in2/? r 1 _
In ~' 22 ~' I 3s ---, I -
r? 2 = -, r 3 - -sin cos e, r? 3 ^ ] , r| 3 -= cotff, r? 4 =
r / lo
(68)
where the accents denote differentiation with respect to r.
Using (68) in (IX. 114) we obtain the tensor R lk from which we get
the scalar R R\ by contraction. Finally, the tensor M lk and the
corresponding mixed components M* are obtained from (4'). A
straightforward consideration shows that the non-diagonal elements of
this tensor are zero and the four diagonal elements M\, M\, M\, M\ are
functions of r only. From the transformation properties of tensor com-
ponents under rotations it then follows that the components M \ and M\
must be equal. However, on account of the identities (11), even the
three quantities M\, M\ = M\ y M\ cannot be independent. As is easily
seen, the equations (11) reduce for a static spherical system to essentially
one equation, only, by means of which the components M \ = M 3 may
be expressed in terms of M\, M\, and dM\/dr. The calculations give the
following expressions for the non-vanishing components of M*.
"""'
Ml- Ml- -- - + + -A.
- -
2a[\b) 2 a b 2\6 / br
These explicit expressions for the components of the tensor M are easily
seen to be in accordance with the identities (11).
XI, 123 THE GENERAL THEORY OF RELATIVITY 325
123. Schwarzschild's exterior solution
In the empty space surrounding a material particle of mass M we
have T* = and the field equations (13) reduce to
a)
1 [76Y la' 6' 1/6V V
r- 2
abr r 2
(706)
v ;
^> (7 C)
the last equation being actually a consequence of the two first equations.
From (70 a) and (706) we get by subtraction
a'b+ab' _ ,
- of~~~ ~ ^
a z br
i.e. (ab)' = (71)
or ab constant. (72)
Putting y - (73)
a
(706) becomes, after multiplication by r 2 ,
ry'+2/-l + Ara = (74)
or (yr)' 1 Ar 2 . (75)
By integration we thus get
A? 3
'iy r _ -- 2m, (76)
o
where m is a constant of integration, or
y =!_*?*!. (77)
y o
From (73) and (77) we get the following solution of (706).
(78)
The spatial line element which in our case is equal to the spatial part
of (63) is thus
If we neglect the small A-term, da 2 will in the limit of large r go over
into the usual line element for a Euclidean space in polar coordinates,
326 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 123
and it is interesting that the condition of spherical symmetry is sufficient
to secure this result without any explicit use of boundary conditions at
infinity. This result is, of course, also partly connected with the normaliza-
tion (60) of the variable r which has been chosen so that the geometry
on a surface r constant is the same as on a sphere of radius r in a
Euclidean space. In the actual space (79), r will, however, not simply
be the radial distance, the distance between two points (r^d^) and
(r 2 , 0, (f>) measured with standard measuring-rods being
I = J (1 2m/r Ar 2 /3)~* dr. % (80)
An observer at great distance from the central body will, however,
try to make a picture of the system in a Euclidean space ; in this
picture the quantity r plays the role of the distance from the centre, and
the distance / measured by standard measuring-rods has 110 real impor-
tance in astronomy. The coordinates (r, 0, (/>) may therefore be taken to
be the usual polar coordinates applied in celestial mechanics.
According to (72) we have
, constant , , /. 2m Ar 2 \ /01X
\) =_ _ -^ constant X 1 . (81)
a \ r 3;
By a simple change of the time-scale the constant may be made equal
to 1, and we thus arrive at Schwarzschild's exterior solution |
/ 2 _ ^ /2 I .2/,/#2 I 2fl 7J,2\ /] ^W ^
\ 2m/r Ar 2 /3 \ r
(82)
or, if we neglect the A- term which is of importance only for very large
values of r,
- dt*. (83)
1 2m IT \ r I
This expression must hold outside a spherical distribution of matter
and, since it is singular at a distance r = r determined by the equation
1-^ = 0, (84)
r o
we may conclude that the 'radius' of the mass must be larger than the
value determined by (84). The singularity at r = r cannot be completely
removed by the use of other 'static' coordinates. It can, however, be
f K. Schwarzschild, Berl. Ber , p. 189 (1916)
XI, 123 THE GENERAL THEORY OF RELATIVITY 327
modified, for instance, by using 'isotropic' coordinates r', 6, <, t defined
by the transformation
r - r ' l + -p' r> = \{(r*-2mr)*+r~m}. (85)
The line element then takes the form ^
(86)
The singular point r = 2m has the new radial coordinate r Q = |m, and
it is seen that in (86) the singularity has been removed from the spatial
part of the line element, but it appears again in (/ 44 which vanishes at
the same place. As shown by Sermi, Einstein, and Pauli,f no non-
singular solutions of the field equations for empty space exist which are
stationary and have the form g^ 1 ^constant/r at infinity. As we
shall see presently, this form of </ 44 indicates that the field is produced
by a mass distribution around the origin. The scalar potential which, of
course, is invariant under the transformation (85) has the following forms
in the two systems of coordinates corresponding to (83) and (86)
, , c 2 me 2 \
X "= (-1/44- 0^ -- -
c 2 we* ' (8?)
X - (-^4-1)3- = ~/ (H l /2 /)2J
At large distances where the field is weak, both expressions reduce to
the Newtonian form mc' 2 /r me 2 //', which shows that the constant
m must be connected with the mass M of the particle creating the field
by the relation i 1/f <> -*
,/ L-> vl ifC* VI
m = ^^ KCjXl -. (88)
C" 877
In the regions where the field is weak, (86) reduces to the expression
where we have put
x r'sin#cos0, ?/ r'sin#sin</, z ~ /'cos^. (86")
On account of (88) this expression is identical with the static solution
(40) of the linear approximate field equations (30)
As first remarked by Lernaitre, J the singularity disappears in the line
t R Senni, Atti Accad Lmcei (5), 27 1 , 235 (1918), A Einstein, Revista (Unw Nac.
Tncuman), A, 2, 11 (1941), A Einstein and W Pauh, Ann. oj Math. 44, 131 (1943).
J (i E Lemaitre, Ann *S 1 oc. Scient. Bruxellex, S'*i A, 53, 51 (1933), see also J L.
Syngo, Proc Roy Insh Soc 53, No 6, 83 (1950)
328 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 123
element if we introduce a non-static system of coordinates r', 0, <f>, t' by
the transformations
r = (9m/2)i(r'-cO, dt' = dt~ (2m / r) * dr. (89)
1 2m/ r
The line element (83) then takes the form
ds* = ^dr' 2 +r 2 (d0 2 +sin 2 rf< 2 )-c 2 eft' 2 , (90)
where r depends on r' and Z' by (89). The new system of coordinates is
of the type considered in 113, where the dynamical action of the
gravitational field has been transformed away, the dynamical potentials
(y t , x) being zero. The motion of a planet is described in these co-
ordinates by the equations (X. 20'), i e. the covariant components of
the momentum vector of the particle at the time t'+dt' are obtained from
the corresponding vector at the time t 1 by parallel displacement in the
space with the line element
9m
da* - dr' 2 +r 2 (d0 a +sin 2 0d^ 2 ).
On account of the time-dependence of r the system of reference is not
rigid, however, and the motion of the planet in these coordinates is
therefore rather complicated.
124. Schwarzschild's solution for the interior of a perfect fluid
The energy-momentum tensor of a perfect fluid is given by (X. 48)
. (91)
On account of the field equations (13),
M\ = icTf, (92)
and the expressions (69) for the tensor Jf J, we see that a static spherically
symmetric field of the type (63) is possible only if the velocity u l of the
matter is zero and if the proper mass density /t and the proper pressure $
are functions of r only. Hence we have, according to (X. 2, 3) and (65),
(92')
XI, 124 THE GENERAL THEORY OF RELATIVITY 329
Using this with the conservation law
1 d(J\a\T k \
mk * ^VV I" I 1 / T*r mis A /QQ\
J. * If ; J. . _ J[ -, \J , I 7 O I
' llftl f^fn
we get for i = 1
1
-ri,* = o
or, by means of (68) and (IX. 69),
This equation gives the dependence of the pressure on the scalar gravita-
tional potential in the equilibrium state of a fluid under the influence
of its own gravitational field. The other three equations of the con-
servation laws (93) do not give anything new.
The field equations (92) again reduce to only two independent equa-
tions for which we may take the equations
M\ - - K T\, Mi - -KTl (95)
the other equations (92) being consequences of (95) on account of the
conservation equation (94). From (69 a, 6), (92'), and (95) we thus get
~\ll --}+* = <$, (96a)
abr r 2 a
The equations (94), (96) together with the equation of state of the
matter which gives the connexion between p and /St determine the
interior state and the gravitational field of the fluid.
For simplicity we shall assume that the fluid is practically incom-
pressible. The proper mass density may then be treated as a constant
and the solution of (966) may be obtained from the solution (78) of
(706) by the substitution A->A+^c 2 . Since our solution is to be
regular as r->0, the constant of integration 2m in (78) must be put
equal to zero. Hence we get
1 1
1 _ ^ r r 2 1
Q J?2
where JP = ^-,. (98)
330 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 124
Further, on account of the constancy of /t, we get at once by integration
of (94) (/ic 2 + J p)v / 6 - constant.
By addition of the equations (96 a, 966) and multiplication by V6, we
therefore have , , , /,
-7, H o constant.
a\or a*r
Substituting in this equation the expression (97) for a we obtain
^WL^I^^A, (99)
r dr
where A is a constant. Putting y = V6 and introducing a new variable
x (1 r 2 /# 2 )* instead of r, (99) may also be written
y-xfy = A. (100)
(ji/JC
The solution of (100) is y = A Bx,
where B is a constant of integration.
Hence b - ?/ 2 -- (.4 B<J(\- r 2 /fl 2 )) 2 . (101 )
Finally, using (97) and (101) with (96 a), we get the following expression
for the pressure p measured in a local system of inertia
Thus we arrive at Schwarzschild's interior solution f
2
ds* =
(103)
The spatial geometry is defined by the line element
= 8 ). (104)
Hence, the geometry on the surface r ? l =^ constant is the same as on a
sphere of radius r l in Euclidean space, but r l is not the distance to the
origin r = measured by standard rods, this distance being
(105)
t K. Schuarzschild, Bert Her , p 424(1916).
XI, 124 THE GENERAL THEORY OF RELATIVITY 331
t
The volume of this sphere is
r\ TT 2rr
Consider a sphere of fluid filling the space inside r r with a
constant proper density of mass /2,. For r < r 1 we then have the solution
(103), while Schwarzschild's exterior solution (82) must be valid for
r > r x . We now have to adjust the constants A and B so that (103)
and (82) coincide for r r v further, p has to be zero at the surface of the
sphere. If we neglect the A-term, which anyhow has only a small effect
inside the solar system, these conditions lead to the equations
with the solutions
A = i
(107)
A comparison with (88) shows that the gravitational field of the
spherical fluid at great distances corresponds to a mass
M = ^ r r?A , (108)
which is thus the same as for a constant distribution of Newtonian mass
over a sphere of radius r l in a Euclidean space.
According to (106) the real volume of the sphere is larger than ~-r\ ;
5
on the other hand, /l was the mass density measured in a local system of
inertia which differs from the mass density in the system of coordinates
used here. Actually we shall show in 128 that the quantity M is exactly
equal to the total energy of the system divided by c 2 . Anyhow, the
difference between the real volume V l given by (106) and ^-rl is in all
astronomical applications very small. In the case of the sun, for instance,
we may put
/t = l-4gm. cm.- 3 , r l = 6-95x 10 10 cm. (109)
332 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 124
Hence we get
= 3-5X
2X10- 3
H
(110)
Also the difference between the distance Z x defined by (105) and the
radial coordinate r x is far too small to be detected by the astronomical
determination of r v
We also see that the condition r l > r 2m for the applicability of
the exterior solution (83) is amply satisfied; for from (107) we get
2m /? .. A
... 1 r+-> lfj~6 ^ 1
' - -?& r^ 1 U <. 1 .
r l E 2
As in the case of empty space we can also here introduce isotropic
coordinates. For arbitrary functions a(r), b(r) in (63), this may be
obtained by a transformation r' = r f (r) satisfying the differential equation
^- = VWy, (111)
the general solution of which is
(112)
where C is an arbitrary constant. The line element then takes the form
(113)
= L. (dx*+dy*+dz*) - 6c 2 dt*
where (x,y,z) are given by (86").
For a(r) of the form (97) we thus get
and the line element inside an incompressible fluid takes the form
The constant C can be determined so as to make (114) coincide with (85) at
the boundary of the fluid.
XI, 124 THE GENERAL THEORY OF RELATIVITY 333
Schwarzschild's solution represents the only exact solution of the
gravitational field equations which has found any application in astro-
nomy. Reissnerf and WeylJ also solved the problem of the* gravitational
field produced by the electromagnetic energy in the surroundings of a
charged particle. The result obtained by these authors is a line element of
the form
l~2m/r+Ke 2 /r 2 '
(116)
The ratio of the two terms depending on the charge and the. mass,
respectively, is thus, on account of (88),
K6 2 477-e 2
r.2m rMc 2 '
(111)
In the case of an electron, the two terms will therefore become of the
same order of magnitude at a distance corresponding to the classical
electron radius a = e 2 /Jfc 2 . Both terms will, however, have negligible
effects on the interaction between electrons as compared with the
Coulomb interaction.
Further, exact solutions of the field equations for the case of arbitrary
cylindrically symmetrical distributions of matter were given by Weyl
and by Levi-Civita.||
125. The variational principle for gravitational fields
Let be an arbitrary domain in 4-space. Consider the four-dimen-
sional invariant integral
J l= = ( RdZ= ( R lk g* k <J(g) dx l dx*dx*dx\ (118)
s E
where R R lk g lk is the curvature scalar and
is the contracted curvature tensor. The integrand in ( 1 1 8) is an algebraic
function of the g ik and their derivatives. Since the contra variant com-
ponents g lk are uniquely determined by the g ik , the integrand can also be
expressed as a function of the g lk and their derivatives. Let us now con-
sider a variation of the metric tensor 8g lk which is arbitrary inside S,
t H. Reissner, Ann. d. Phys. 50, 106 (1916).
t H. Weyl, ibid. 54, 117 (1917) ; Raum-Zeit-Materie, 3rd ed. f p. 223, Berlin, 1920.
H. Weyl, Ann. d. Phya. 54, 117 (1917); 59, 185 (1919).
|| T. Levi-Civita, Rend. Accod. Lincei (5), No. 26 (1917), No. 27 (1918); No. 28 (1919).
334 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 125
but which vanishes together with the variation of the first derivatives
at the boundary of S. The corresponding variation of J x is then
S/! - J %R ik J(~g)g lk dx+f R lk S(V(-W) dx. (120)
By variation of (119) we get
(121)
While the Christoffel symbols T\ t transform according to the trans-
formation equations (IX. 53) which, on account of the first term on the
right-hand side, differ from the transformation law of tensors, the varia-
tion SFJ.,, which is the difference between two Christoffel symbols at the
same point, will obviously transform like a tensor. Thus, the covariant
derivative
( 8r ). = ^J+ r*mr sr&-rj w sr;,-rr ffl 8r; r (122)
will also be a tensor. This allows us to write the tensor 8^ ?fc in the simple
form fc -(sri*).,. (123)
This tensor relation is most easily proved by introducing a geodesic
system of coordinates in which FJ. Z 0; for in this system the right-
hand sides of (121) and (123) are at once seen to be identical.
If we multiply (123) by ^(~g)g lk we get, using (IX. 85, 86, 90),
Hence the integrand of the first integral in (120) has the form of an
ordinary divergence. This integral can therefore be transformed to an
integral over the surface of 2 and, as the variations 8g lk and their first
derivatives are zero on this surface, the first integral in (120) is zero.
Further, we get from (IX. 69')
and the integrand of the second integral in (120) becomes
XI, 12.5 THE GENERAL THEORY OF RELATIVITY 335
Thus we get for the variation of the invariant J^
80'V(-flO ** ( 125 )
Similarly, we obtain by means of (124)
8 J 2AV(-</) dx - - J \g lk S/'V(-ff) ** < 12 )
Hence, adding (125) and (126), we see that the variation of the invariant
flf)cfa (127)
is equal to 8J =-= [ Jl/^ $y lk <J(g) dx, (128)
where 37, A is the tensor appearing on the left-hand side of the field equa-
tions (12). The field equations in empty space
M lk = (129)
are therefore equivalent to the vanational principle
5.7 - (130)
for all variations where 8f/' A and their first derivatives vanish at the
boundary of X.
Tt is clear that (125) and (128) remain true if from J and J we
subtract any integral whose integrand has the form of an ordinary
divergence, for this can be transformed into a surface integral which
will give no contribution to 8^ and S.7 for the variations in question. Now
the contribution to R<J(y) - R lk y lK J(g) arising from the first two
terms in (110) can be written
and, since the first two terms are ordinary divergences, they may thus be
neglected. Further, substituting for &g lk /dx? the expression (IX. 68) in
terms of the Ohristoffel symbols, and using the relation
the last two terms in (131) are easity seen to reduce to 2 with
s ^ j(-g)g<(v; k ri rl - r^rL). (132)
336 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 125
Since the contribution to R^J(-g) from the last two terms in (119) is
fi, we get from (125) and (128)
8 J fi dx - J J(~g)(R lk -\Rg lk ) W k dx (133)
and 8 J dx = j *J(-g)M lk 8g* k dx, (134)
where = Z+2Xj(-g). (135)
The integrals J dx and J dx are not invariants. Nevertheless, since
they are defined by the same equations (132), (135) in every system of
coordinates, the relations (133) and (134) have a co variant meaning.
In contrast to the integrands of J a and J, fi and do not contain the
second derivatives of the metrical tensor. They can therefore be regarded
as functions of the quantities g lk and their first derivatives
(136)
o
As 8g] k = ,$g lk , we obviously have
dx 1
= ff *--,(-
J [8gT< daf\d
and similarly
1 "*
Since the expressions (133) and (137) for 8 J fi dx must be equal for
any variation 8g lk inside the arbitrary domain S, we must have at every
In the same way we get by comparison of (134) arid (138)
In applying this argument we have treated the variables g lk , g\ k as inde-
pendent, thus disregarding the symmetry relations g lk = g kl , g\ k = gf 1 .
This is obviously permissible, provided that the result of the derivations
on the right-hand sides of (139) and (140) is symmetrical in i and k.
Although and $ in general have rather complicated transformation
properties, the covariant expressions on the right-hand sides of (139) and
(140) must transform like tensor densities.
XI, 125 THE GENERAL THEORY OF RELATIVITY 337
If for constant g\ k all the variables y lk are multiplied by a factor A,
the quantities g lk> <J(g), and Fj^ are from (IX. 7, 4, 50) seen to be
multiplied by the factors A" 1 , A~ 2 , A~ l , respectively. Hence, the quantity
2 will be multiplied by the factor A~ 3 , i.e. 2 is a- homogeneous function of
the g lk of degree 3. From Euler's theorem we thus get the equation
In the same way we see that as a function of the variables g\ k is
homogeneous of degree 2. Hence we also have
126. The laws of conservation of energy and momentum
In 115 it was shown that the conservation of electric charge is a
consequence of the covariant divergence relation div{$ 1 } 0. This is
connected with the circumstance that the vanishing of a covariant diver-
gence of a four-vector is equivalent to the vanishing of an ordinary
divergence of a vector density. By subsequent integration over the
space coordinates we are then at once led to the conclusion that the
total charge is constant in time.
The law of conservation of energy and momentum, which has the form
(X.41)or
(143)
is, however, not in general equivalent to the vanishing of an ordinary
divergence and will therefore not immediately give rise to any conserva-
tion laws by integration over the space coordinates. Only in the case
of a stationary system considered m 114 is the right-hand side of
(143) zero for i = 4, and by subsequent integration over the space
coordinates we get a constant of the motion which may be interpreted
as the total energy.
The occurrence of the term on the right-hand side of (143) indicates
that the system is not strictly closed, this term being analogous to the
external four-force density on a non-closed system in the special theory
of relativity (cf. Chapter VII). In the case of electromagnetic forces, it
was possible by means of Maxwell's equations to write the four-force
density as the divergence of the electromagnetic energy-momentum
tensor. Similarly, by virtue of the field equations (12), the term
3595.60
338 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 126
on the right-hand side of (143) can be written in a covariant way in the
._.
dx"
where t k is a scheme of 4 2 quantities depending on the components of the
metric tensor and their first derivatives. The quantities k % and t* will
of course not transform like tensors, but the equations (143), (144),
(145) will, as we shall see, by integrations over the space coordinates,
give rise to conservation theorems for quantities which have simple
transformation properties and a simple physical meaning. f
In order to prove (145) we first remark that (144) can also be written
by virtue of the rules for lowering indices together with the relations
(IX. 7).
Next we substitute from (12) in (144') and use (140), which gives
As is a function of g lm and tfg 1 only and
the sum of the first two terms inside the brackets is equal to
Thus we see that k ( is of the form (145) with
(146)
Using (145) in (143) the laws of conservation of energy and momentum
can be written in the form
where X { * = j(- g )(T*+t*). (148)
t A. Einstein, Berl. Ber., p. 778 (1915), Ann. d Phys. 49, 769 (1916), Perl. Ber.,
p. 448 (1918), F. Klein, Odtt. Nachr., Math.-phys. Klaase, p 394 (1918).
XI, 126 THE GENERAL THEORY OF RELATIVITY 339
The quantities t t k /\l( g) do not in general transform like a tensor.
Since is quadratic in y[ m , the quantities t* can, if we neglect the A-
terms, even be made equal to zero at any given point by introducing a
system of coordinates which is geodesic at this point. Z l k /^(g) will
behave like a tensor only if the transformation coefficients af are
constant. However, by integration over the spatial coordinates we
can derive a set of quantities which behave like a four-vector under
a much wider group of transformations. Let us consider an isolated
system for which the TJ- are different from zero only inside a certain
tube with a finite space-like extension in 4-space. For sufficiently
large space-like distances from the tube, we may assume that for a proper
choice of coordinates the y lk reduce to the constant values G lk of the
special theory of relativity. Such coordinates, which can be quite
arbitrary inside the tube, will be called quasi-Galilean. If we neglect
the A-term, the Z t k will therefore be zero at sufficiently large spatial
distances from our system, and by integration of (147) over the space
coordinates x l , x 2 , x* we get at once
d f 3; * dx*dx*dx* - 0. (149)
dx* J
Hence we see that the quantities
P l = X* dx*dx*dx* = c l + \ W+'i 4 ) dV (150)
are constant in time.
Further, it is easily seen that the PI are invariant for any trans-
formation of coordinates x' 1 = x' l (x k ) inside the tube which leaves the
x l unchanged outside the tube. To prove this we only need to introduce
a third system of coordinates x" 1 which on the hypersurface iij_ defined
by x"* = a = constant coincides with the system (x 1 ) and on the different
hypersurface 13 2 , where x"* b, coincides with the system x' 1 . Since
the constancy of the quantities (150) holds in all three systems of
coordinates we get at once
On the other hand, for linear orthogonal transformations
x' 1 = oclx k (152)
with a determinant = |o*| = 1, the quantities P t will transform like
the covariant components of a four- vector in a pseudo- Cartesian system
of coordinates. For the proof, we first remark that the quantities
X T k transform like the mixed components of a tensor under the
340 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 126
transformations (152) with constant coefficients a*. Now consider a
four-vector a 1 whose components are constant inside the tube. The
components a , t = ^ (153)
are then also constant inside the tube, and the quantities
b k = Xfa 1 (154)
will, on account of (147), satisfy the equations
in all systems of coordinates connected by the linear orthogonal trans-
formations (152). Thus, formally, the situation here is the same as in
63, apart from the use of the imaginary time representation in Chapter
VI. By the same reasoning as that used for the derivation of (VI. 31),
it now follows from (155) that the quantity (b*lc)dx*dx 2 dx* =~ P l a 1 is
invariant under the orthogonal transformations (152) and, as this must
hold for an arbitrary constant vector a 1 , the integrals PI must transform
like the covariant components of a four-vector. Combining this result
with the fact expressed by (151) we see that the P l must behave like a
four-vector by any transformation which outside the tube has the form
of a Lorentz transformation.
The quantities P i (P t , Ejc) represent the total momentum P and
energy E of an isolated system. If t* = 0, the expression for the total
energy E reduces to the expression obtained in 114 for a stationary
system. The last term in the integrals (150) may be interpreted as the
contribution of the gravitational field to the total momentum and
energy. As already mentioned in 112, a unique separation into a
material and a gravitational part is not, however, possible. The separa-
tion depends, m general, on the coordinates used in the evaluation of
the energy and momentum.
It is now tempting to consider (147) as the differential form of the con-
servation laws. The quantities
(156)
c
which in the case of if reduce to the expressions (X. 45, 47), will
then be interpreted as the energy flux, energy density, and momentum
XI, 126 THE GENERAL THEORY OF RELATIVITY 341
density, respectively. It should be remarked, however, that this inter-
pretation is not independent of the coordinates used; for tj and t* and
therefore S l and g L will not in general transform like the components
of space vectors by a simple change of the spatial coordinates, neither
will k in general be an invariant under such transformations. Only the
expressions (150) for the total momentum and energy have an invariant
meaning which is physically satisfactory.
127. Different expressions for the densities of energy and
momentum
By means of (146) and (132) we can now write down explicit expres-
sions for the quantities t t k . As shown in Appendix 9, the derivative of
with respect to g l l is given by
)u- (is?)
After a simple calculation we then get, remembering that
_ i aV(-) _ , W
- ~~ ~ ~
flU*- r* - p - S ** (15*)
-M. - 1 Jm - -- i. -- -- *.$ (158)
As shown by Tolman,")" the quantities Z t k may also be expressed in
the very useful form of an ordinary divergence
From (148), (13), and (146) we get
in which the A-term has disappeared. This expression can be further
transformed by means of (139) :
t R. C. Tolman, Phys. Rev, 35, 875 (1930).
342 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 127
As shown in Appendix 9, the last term is, however, identically zero.
Hence, Xf may be written in the form (159) of an ordinary divergence
of the quantity ' , o p
8* m = ~(7 W . (162)
K vy m
Further, we get from (148), (146), (142), and (159), neglecting the
small A- terms,
-_.= J(- ff )T'_ ~= ' . (163)
vv " ' K dx m
Substituting the expression for obtained from this equation in
the expression (146) we get for t 4 * and, finally, for the 'energy density'
Vx* V
from which the total energy is obtained by integration over the spatial
coordinates.
128. The gravitational mass and total energy and momentum
of an isolated system
By an isolated system we understand a system which allows the
introduction of quasi-Galilean coordinates x l = (x, y, z, ct) in which the
line element at great distance from the system takes the form (86'),
where m is a constant. For the total energy-momentum vector P l of the
system we then get by means of (150) and (159)
= - f I, 4 dx^dx*dx* = ! f ^ dx*d
1 cj l c] dxv
(165)
The first integral on the right-hand side can be transformed into an
integral over an infinitely distant spherical surface
r = ( x *+y*+z 2 )* = constant.
It thus depends only on the values of the metric tensor and its derivatives
at large distances which are time-independent. As the vector PI is
constant for an isolated system, the last integral must also be constant;
XI, 128 THE GENERAL THEORY OF RELATIVITY 343
actually it must have the value zero, fcince the integral involved cannot
change permanently at a constant finite rate in an isolated system.
Hence, we get , /.
P^ = - c \ 8^ da, (166)
where n^ = dr/dx* 1 = (xjr, y/r, z/r) is a unit vectofr in the direction of tha
outward normal, and the integral is extended over the surfac^ of the
infinitely distant sphere r = constant. At large distant^ we have,
however, according to (86'),
.2m /. 2ra\
ffii = 022 =~ 033 = ! + > V" =
all other components being zero. Hence,
S^i.SW.-IU.-UfS. ,,67,
If we neglect terms of the order m/r compared with unity, we can replace
gik jyy Qik anc j ^(~y) \}y \ when multiplied by a Christoffel symbol
F^. From (162) and (157) we therefore get to .this approximation
K
(168)
As G lk =^= for i 7^ k the only Christoffel symbols occurring in this
expression are the following, which aie easily calculated by means of
(EX. 50) and (167):
T> __ x4^^ ps __ _ M _ s \~ m (2 r8 ri* G^ri* o
1 ^4 t "2 /*' l rs I 1 r4/ 2 " r ' ^ s ~" ^ r "~
(169)
*^7>j
Hence we get */* = -~8f - w -- w u , (170)
ACT*" ^
and from (166)
/> = -8 ? \ f ^[('j' + ^ + m^ = -8f !=?. (171)
cj *r 2 Lw \r/ \r/J KC
As PI ~ (P, E/c), we see that the total momentum is zero in the
system of coordinates corresponding to the line element (86') at infinity,
the isolated system is as a whole at rest in this system. For the total
energy we have, on account of (88),
E = -cP 4 = - = Mc\ (172)
K
i.e. the gravitational mass M , as determined by the scalar gravitational
potential at great distances, is connected with the total energy by
344 FUNDAMENTAL LAWS OF GRAVITATION IN XI, 128
Einstein's relation. For a Lorentz transformation of the quasi-Galilean
coordinates the quantities P l transform like a four-vector and in the
transformed system P and E will have the same values as the momentum
and energy of a moving particle in the special theory of relativity.
In the calculation of the total energy just worked out we have
used the formula (159) which expresses the energy as a function of pure
gravitational field quantities, actually only the field variables at infinity
occurring in the expression (166). Sometimes it is convenient to use
instead the formula (164). For a stationary or a quasi-stationary system
where all terms depending on time derivatives can be neglected, we get
E = -cP 4 = - J X 4 4 dx l dx*dx*
J J r'
In the last integral the integration is again extended over the distant
sphere of radius r = constant, hence we get by means of (162), (157),
(169), and (170)
*;/* == -{rft~i(8^rf r +8r^)-j(G rs i> -o^r^o^o 11
i Lf " ir * i ir/ ** \ rs ro/ ID
K
= -^n =aj*,
and a comparison with (166) shows that the last integral in (173) is
equal to E. Thus we get for E the simple expression
T\)dV, (174)
which contains only an integration over the part of space where T* is
different from zero, i.e. where there is actually matter present. A com-
parison of (174) with (172) shows that the quantity
Tl) (175)
may be interpreted as the density of gravitational mass in the system.
Since /* obviously behaves like a scalar for all purely spatial transforma-
tions this interpretation has a well-defined physical meaning.
In the case of an incompressible fluid at rest, treated in 124, we have,
according to (92'),
T\ = T\ = T\ = $, T\ = -/I c 2 .
XI, 128 THE GENERAL THEORY OF RELATIVITY 345
Further, for A = 0, we have on account of (103), (104), (102), (107),
and (175),
rsn
Hence, we get for the total field -producing gravitational mass of the
spherical fluid
in accordance with (108). While the proper rest mass density /S is
constant in an incompressible fluid, the gravitational mass density IJL
decreases with r according to (176).
XII
EXPERIMENTAL VERIFICATION OF THE
GENERAL THEORY OF RELATIVITY.
COSMOLOGICAL PROBLEMS
129. The gravitational shift of spectral lines
WHILE the consequences of the special theory of relativity have been
verified to a very high degree of accuracy by numerous experiments,
the experimental verification of the general theory has so far been
limited to three cases only. The reason for this is obvious and is con-
nected with the fact that the Newtonian gravitational theory represents
a very good approximation for all gravitational phenomena inside the
solar system.
The most elementary of the effects which represent a crucial test of the
general theory of relativity is the gravitational shift of spectral lines,
which is a direct consequence of the principle of equivalence. According
to (VIII. 100), the rate of a standard clock at rest in a gravitational field
at a place with the scalar gravitational potential x is connected with the
rate of the coordinate clock defining the time / of our system of co-
ordinates by the formulaf
dr ^dt(l+2 x /c^ (I)
Now consider an atom at rest at the place p l with the scalar potential
Xi, emitting light of the proper frequency v. Then v is equal to the num-
ber of light waves emitted per unit time in the time-scale of a local rest
system of inertia which is the same as the scale of the standard clock at
rest at the point p r The number of waves emitted per unit time in the
scale of the corresponding coordinate clock will then be
v 1 = $(l+2 Xl /c), (2)
for, according to (1), the time-interval AT of the standard clock corre-
sponding to AJ 1 is AT = (l + 2^ 1 /c 2 ) i . If the gravitational field is
stationary, the (j lk are independent of t and the number of waves reaching
an observer at an arbitrary point p 2 per unit time in the t -scale must be
constant in time and equal to the number of waves v emitted from ^
f This formula, which is a consequence of the principle of equivalence, has also played
an impoitaiit part m discussions between Einstein and Bohr on the consistency of the
quantum mechanical description See the article by N Bohr in Albert Einstein Phtlo-
ftopher-fictenttst, Library of Living Philosophers, voi vn, Kvanston, 1949.
XII, 129 THE GENERAL THEORY OF RELATIVITY 347
per unit time in the same scale. The frequency v of the radiation measured
by a standard clock at rest at p 2 will then be
v = v 1 (l + 2 X2 /c2)-^, (3)
where ^ 2 is the scalar potential at the place p 2 , for a unit time-interval
in the scale of the standard clock at p 2 corresponds to a time-interval
From (2) and (3) we thus get
(*>
Hence the observed frequency v will differ from the proper frequency v
by an amount Ay = v-~v, which in the case of weak fields is given by
where A# = Xi~~Xz * s ^e difference between the potentials at the places
where the light is emitted and observed, respectively.
In the gravitational field of the sun we have, according to (XI. 83, 88),
me 2 kM
where M = 1-983 X 10 33 gm. (5')
is the mass of the sun. Hence we get for the shift of a spectral line
emitted by an atom in the outer layers of the sun as compared with the
same line emitted on the earth
^ =_*(!_.!),
c 2 \r t rj
(6)
where i\ and r 2 are the r -values at the surface of the sun and at the
distance of the earth, respectively. These values of r are not exactly
equal to the values of the distances calculated by means of the usual
astronomical methods which are based on the assumption that the space
is Euclidean and that the light rays are moving in straight lines. The
differences are small, however, and since the effect (6) is itself small, the
error introduced by using the astronomical values of the distances
instead of the correct r- values is small of a higher order. Since r 2 ^ r ly
we get, putting r l equal to the usual astronomical value of the radius
ofthesun > ^ = 6-96x10" cm., (6')
the value ^r = 2 ' 12 X 10~ 6 (7)
v
348 EXPERIMENTAL VERIFICATION OF XII, 129
for the line shift. The observed spectral lines should therefore be shifted
slightly towards the red. This interesting effect predicted by Einstein!
is in satisfactory agreement with the observations both in the case of the
sun and in the case of the heavy companion of Sirius, where the effect is
about thirty times larger 4
130. The advance of the perihelion of Mercury
Let us consider the motion of a particle (a planet) in the gravitational
field of a much heavier body (the sun) described by Schwarzschild's
exterior solution (XI. 83, 88, 24). In this case, the components of the
metric tensor are
ffn = j ,- > 22 = ** 2 , 0r 33 = r 2 sin 2 0, g u = - / 1 - ^J
A f . , f 2kM K Mc*
g lk = for i ^ k, a = =
C 47T
(8)
Hence
Yi = > Yuc = X = -"2
According to (X. 17, 17'), the gravitational force on the particle is
(io)
where m = * T =
is the relativistic mass of the planet. The square of the velocity is
r 2
- , (11)
where the dots mean differentiation with respect to the time-variable t.
Further, by (X. 12) and (8), the momentum vector of the particle is
The equations of motion (X. 15)
_ j
dt dt 2 dx l P '
t A. Einstein,. Ann. d. Phys. 35 r 898 (1911).
I St. John, Astrophy*. Journ. 67, 195 (1928); W. Adams, Proc. Nat. Acad. 11, 382
(1925); see also E. F. Fjeundlich and W. Ledermann, Mon. Not. Rvy. Astr. Soc. 104,
1, 40 (1944).
XII, 130 THE GENERAL THEORY OF RELATIVITY 349
are thus in the present case
= ) (146)
Guv
and ~(rr*sinWd>) = 0. (He)
at
Since the gravitational field is static, the energy H of the particle defined
by (X. 22) is constant. Hence
is a first integral of the equations of motion, where E is a constant of
integration representing the energy of the system divided by ?fe c 2 .
From (146) we see that 01
^ '
represents another integral of the equations of motion. On account of
the central symmetry of our problem, any plane through the centre may,
however, be chosen as the plane 9 |TT, i.e. the orbit can lie in any plane
through the centre. Substituting from (16) in (14c) we get at once the
further integral ^ _ ^ (17)
where C is another constant of integration. On account of (15) this
equation can also be written
(7. (18)
loc/r
Now even for Mercury, the planet nearest to the sun, the quantity
a/r is a very small quantity of the order of magnitude
OL iJjJL i /-v Q / t f\\
- = , 5X10- 8 , (19)
r c 2 r
and for the other planets a/r is much smaller. Therefore, in all actual
cases, the gravitational field may be treated as a weak field and, since
also u z /c 2 <^ 1 , we may to a first approximation use the approximation
(X. 25) for H in (15). Further, using (11) and (16) we therefore get for
(15) to this approximation
- = constant, (20)
350 EXPERIMENTAL VERIFICATION OF XII, 130
which is the usual energy equation in Newton's theory. To the same
approximation, (18) reduces to
r 2 <f> = C, (21)
expressing the conservation of angular momentum. The orbit deter-
mined by (20) and (21) is therefore an ellipse
r _? ? /22)
where is the eccentricity, and
are the r- values corresponding to aphelion and perihelion of the planet,
respectively. For Mercury we have
e 0-2056, a = 5-786 X 10 12 cm. (24)
For stronger fields the equation (20) has to be replaced by the energy
equation (15), and instead of (21) we have the integral (18). However,
the left-hand side of (18) cannot in general be interpreted as angular
momentum, since the notion of a 'radius vector' occurring in the defini-
tion of the angular momentum has an unambiguous meaning only in a
Euclidean space.
In order to determine the orbit m the general case, we introduce the
quantity p l/r instead of r. Thus we get from (18)
dr drdCI a\ ldp\n/-i \ /or\
- = - (l -" p) ' (28)
and by means of (11), (16), (18), and (25) we get for the velocity of the
particle r/r/ \ 2 I
* - G( 1 -acp) + p 2 -/> 3 . (26)
Using the expression ( J 0) for F and squaring the energy equation (15) we
get
(27)
Thus, substituting the expression (26) for u 2 and solving with respect to
(dp/dffi, we get the differential equation for the orbit of the particle in
the form 7/7x2
= Al Bp-p'+ap*. (28)
where A and B are constants. On account of (19) the last term in this
equation is very small compared with the term p 2 and, if we neglect it
XII, 130 THE GENERAL THEORY OF RELATIVITY 351
entirely, (28) reduces to the equation obtained from the Newtonian
equations (20), (21), i e.
' (2 >
It is easily seen that (22) is a solution of this approximate equation. The
maximum and minimum values of p corresponding to perihelion and
aphelion are obtained as roots of the quadratic equation
A + Bp-p* = Q. (30)
Let p l and p 2 be the two roots which must be real and positive in the case
of an orbit which stays inside a finite region in space. Then we have,
according to (23),
The equation (29) may now be written
p 1 )(p t -p)}, (32)
and the increase </> 2 cf) l in <f> during an increase of p from p l to p 2 is
obtained by integration
P2
,-*-J
- ^ sin-
(33)
-P)} 2(P2 Pl)
Pl
in accordance with the solution (22).
Returning now to the exact equation (28), we see that the right-hand
side of (28) is a polynomial of third degree and the equation obtained by
putting dp/d(f) has three roots p lt p 2 , p 3 . For small values of a two
of these roots, p l and p 2 say, must be approximately equal to the two
roots (31) of the equation (30), and since
Pi+P2+Pa = - ( 34 )
p 3 must be very large for small values of a. The roots p 1 and p 2 will there-
fore again represent the minimum and maximum values of p in the orbit
of the planet. Instead of (32) we now have
-^ = VHP PI)(P P 2 )(P Pa)}
L 1 a (Pl+P2)JI
where we have used the equation
352 EXPERIMENTAL VERIFICATION OF XII, 130
following from (34). Since ocp and cx(/o 1 +p 2 ) are small quantities we thus
get to a first approximation in OL
*p\
+ 2
and the increase in <f> for an increase of p from p i to /> 2 is
1
J J
PI
' 2 J
X
Hence the difference in <f> between two successive perihelions is
(36)
which differs from the value 2-77 obtained from (33) by the amount
A<=~(p 1 +? 2 ). (37)
This result may be interpreted by assigning to each revolution of the
planet an advance of its perihelion of the amount (37). Since it is a small
effect we can substitute the approximate values (31) for p l and /> 2 - F r
Mercury we get, using (24), an advance of the perihelion of an angle
42'9" per century, which is in satisfactory agreement with the observed
advance after subtraction of the effect due to the perturbation by the
other planets. f For the other planets the advance of perihelion following
from Einstein's theory of gravitation is too small to be observed with
certainty.
Instead of the Schwarzschild form (XI. 83) of the line element we
could in these calculations also have used the form (XI. 86) correspond-
ing to the 'isotropic' coordinates (XI. 85). In these coordinates the
equations of motion are somewhat more complicated, but the final
result regarding the advance of perihelion is, of course, the same
a&that given by (37).
t J. Chazy, CM. 182, 1134 (1926).
XII, 131 THE GENERAL THEORY OF RELATIVITY 353
131. The gravitational deflexion of light
The third of the effects which furnishes a crucial test of the general
theory of relativity is the deflexion of a light ray in the gravitational
field of the sun. Since this field is static, the trajectory of a light ray is,
according to the considerations of 117, determined by Fermat's prin-
ciple, i e. by the equations (X. 100, 99, 94):
- (38)
lK >~~ ~ ~'
dX"d\
a- 1 -
In the field described by (8) and (9) we thus have
(40)
and the two equations (38) with t -~ 2, 3 reduce to
~ (r*6) - V . 2r 2 sin cos f 2 -- 0, -f- (r 2 sin 2 0)^=0. (42)
d\ " d\
From (42) we see again that
B - \TT, r*</> =, C (43)
are integrals and (41) then reduces to
2 * I4A\
44
2/1 a \ - , 2 C * a l
2 1 = r 2 + - =--.
\ r/ r 2 r 3 c 2
Introducing p 1/r as a new variable in (44) we get, since
dX d<f> d\
2 I 3 2 I 3 I A K\
where we have put A cC
Now consider a light ray coming from infinity (p = 0) along the
direction </> = 0. Neglecting first the small term ap 3 in (45), this equation
is easily integrated. We get
p
i.e. /> = --sm<, r = - -. (46)
3595.60 Aa
354 EXPERIMENTAL VERIFICATION OF XII, 131
If we make a picture of the trajectory in a Euclidean plane, where the
coordinates r, <f> are pictured as ordinary polar coordinates, the curve
(46) represents a straight line which passes the centre r at a distance
A for <f> = \TT and which goes to infinity again for </> -> TT.
The exact equation (45) may now be written
where we have introduced a new variable
a = Ap(l ap)* (48)
instead of p. Since ap is a small quantity, we may write to the first order
in a
I /v /" \ I /vnl I rurr\
(49)
Hence A dp = dff(l + ~
and from (47)
According to (47) the maximum value p m of p, i.e. the value for closest
approach of the ray to the sun, corresponds to o- = 1. The corresponding
value of the angle d> is
Thus, in the Euclidean picture mentioned above, the curve (50) is
represented by a slightly curved line with a total deflexion
This will be the deflexion of a light ray in the field of the sun as noted
by an observer on the earth, which in this connexion may be regarded as
infinitely far from the sun. From (49) we get for the p- value correspond-
ing to closest approach
Hence, neglecting terms of second order in a, we get for (52)
(53)
For a light ray which grazes the limb of the sun we then get, by means
of (5') and (6'), an angle of deflexion of 1*1 5". This effect predicted by
XII, 131 THE GENERAL THEORY OF RELATIVITY 355
Einsteinf has been tested by observations during total eclipses of the
sun on the apparent positions of stars whose light has passed close to the
limb of the sun. The agreement between Einstein's formula (52) and
the observations seems to be satisfactory, J but, as the effect is just
inside the limits of experimental error, one cannot attach too much
weight to the quantitative agreement.
The deflexion (52) is due partly to the varying velocity of light ex-
pressed by (40) and partly to the non-Euclidean character of the spatial
geometry. If we had taken the space to be Euclidean, the equation
(44) would have been replaced by
(54)
c 2 (l ct/r)
Instead of (45) we would then have obtained
df)
which by integration to the first order in a leads to a deflexion
i/j = a/A (55)
of only half of the value (52).
On the other hand, if the velocity of light is put equal to the constant c,
while the spatial geometry is taken to be that given by the metric tensor
y iK defined by (9) and (8), the trajectory of the light ray is, according to
(38), a 'straightest' line. In this case the equations (44) and (45) are
replaced by
? (f -"-">(*-
1oL/r
respectively. Integration of the last equation then again leads to a
deflexion / /A ^A\
ifj = a //\ (56)
to the first order in a. Adding the two effects (55) and (56) we thus come
back to Einstein's formula (52).
The direct experimental verifications of the general theory of rela-
tivity are thus very few; it should be remembered, however, that the
general theory is not only a natural, but a nearly cogent generalization of
the experimentally well-founded special theory. Further, since Einstein's
gravitational theory contains Newton's theory as a first approxima-
tion, all the numerous astronomical observations which confirm the
predictions of Newton's theory may therefore in a certain sense also
t A. Einstein, BerL Ber., p. 831 (1915); Ann. d. Phys. 49, 769, 22 (1916).
J W. W. Campbell and R Trumpler, Lick Observ. Bull. 11, 41 (1923); 13, 130 (1928).
356 EXPERIMENTAL VERIFICATION OF XII, 131
be regarded as a support of the general theory of relativity. The fact
that the differences between the two theories become apparent only in
the three small effects discussed above simply shows that Newton's
theory represents an extremely good approximation for all gravitational
phenomena inside the solar system. However, as regards cosmological
problems which concern the structure and motion of larger parts of the
universe, the two theories of gravitation can be expected to lead to very
different results and, on account of its inner consistency and the gener-
ality of its basic principles, Einstein's gravitational theory may be
expected to provide a safer guidance in the handling of these difficult
problems.
132. Cosmological models
It has been known for a long timef that Newton's gravitational
theory meets with serious difficulties when applied to the universe as a
whole. Since Einstein's gravitational theory can be expected to give
results which deviate appreciably from Newton's theory just for systems
of cosmological extension it is clearly of great interest to investigate the
new possibilities for a treatment of the universe as a whole, offered by
the general theory of relativity. This question was taken up by Einstein J
shortly after the development of the general theory and has since then
been the object of many investigations by numerous authors. We shall
not attempt here to give a detailed account of all these investigations,
but confine ourselves to the consideration of the static homogeneous
models of the universe originally proposed by Einstein and by de Sitter. ||
All models of the universe which have been considered so far are
based on the assumption that the world, when looked at from a large-
scale point of view, is spatially homogeneous and isotropic. It is true
that the matter in the universe is partly gathered into stars which again
have a tendency to cluster into nebulae of the same character as our own
galaxy of stars. But, in the portion of space which can be reached by
means of the present telescopes, these nebulae seem on the whole to be
fairly uniformly distributed, and the assumption that the large-scale
properties of the universe can be properly described by treating the
matter as a perfect homogeneous fluid seems to be a natural starting-
point. In the models proposed by Einstein and de Sitter the universe is
t C. Neumann, Kgl sacks Ge# d Wiss zu Leipzig, Math.-nat. KL 26, 97 (1874),
H. v Seehger, Astron. Nachi. 137, p. 129 (1895), Munch. Ber. 26, 373 (1896).
t A. Einstein, Berl Ber , p. 142 (1917), Ann. d. Phys 55, 241 (1918)
A Einstein, Serf Ber., p 142(1917).
|| W. de Sitter, Amst. Proc. 19, 1217 (1917); 20, 229 (1917).
XII, 132 THE GENERAL THEORY OF RELATIVITY 357
furthermore assumed to be a static system, which means that we can
introduce a system of coordinates x l -^ (r t O,<f>,ct) in which the line
element has the static and spherically symmetric form (XI. 63)
rf* 2 -- a(r) (J/- 2 + r 2 (rf0M- m 2 d<f>*)-b(r)c* dt* 9 (57)
where a and b are functions of r only. On account of the assumed homo-
geneity of the universe any point in space may be taken as the origin
r =- of the spatial system of coordinates
The functions a(r) and b(r) are now connected with the proper mass
density /I and the proper pressure j8 in the universe by the field equations
(XI. 96, 94) for a perfect fluid,
'+ = 0, (58)
---
abr r*
ar
Here, p and /t are constants on account of the assumed homogeneity of
the model and we shall look for the possible regular solutions of those
equations.
Since dp/dt 0, (58) reduces to
(/ic 2 -/})// - 0, (60)
which requires either b' - - (61)
or /ir 2 \ fj (62)
These two alternatives lead to tlie solutions of Einstein and de Sitter,
respectively .f
133. The Einstein universe
The solution characterized by the condition (61) corresponds to a
constant value of fe, and by a suitable choice of the time variable / this
constant may be made equal to 1 , i e
6-1. (63)
Introducing (61) into (59 a) arid solving this equation with respect to a,
we obtain , ,
a ^ ~ 2 = ' (64)
f R. C Tolman, Proc Nat Acad 15, 297 (1929)
358 EXPERIMENTAL VERIFICATION OF XII, 133
where we have introduced a new constant E by the equation
Further, using (64) and (596), we get
(65)
-p = l*(/i e+$), (66)
which together with (65) leads to the relation
A - i*(Ac 2 +3$). (67)
Now the quantity /3, is essentially positive and even if we allow for
possible cohesive forces in the fluid giving rise to a negative value of ft,
the order of magnitude of p will for any reasonable properties of the
matter be far below the value /lc 2 . Hence the constants A and l/E 2
must also be positive and B will therefore be a real quantity of the dimen-
sion of a length.
In his original paper Einstein assumed the matter in the universe to
be incoherent, thus exerting no pressure at all. In this case, we get from
(66) and (67) 2
^i-T*'- (68)
On the other hand, if the universe is assumed to be mainly filled with
radiation, we have, according to (VII. 145),
which leads to the relation
3 1
> ^,,0/2 f7{\\
A== 2^ = ^ C ' (70)
According to the estimate of Hubble,f the lower limit for the mean
density of matter in the actual universe is of the order
fr tt 10- 30 gm./cm. 3
From (68) and (XI. 24) we then get
:= _L~ 9xlO~ 58 cm.~ 2 ,
(71)
and, as an upper limit for E, we obtain
E 3 X 1C 28 cm. ^ 3 X 10 10 light years
f E Hubble, Astrophys. Journ. 79, 8 (1934).
XII, 133 THE GENERAL THEORY OF RELATIVITY 359
According to (57), (63), and (64) the line element of the Einstein
universe is 7 2
In a region of space for which
^<!> (73)
this line element reduces to that of the special theory of relativity.
With the value of A and R given by (71) we get for the distance r tt IO 14
of the planet Neptune
T 2 /]0 14 \ 2
:__ ~ [ [ = io- 28 ,
\ /
which means that the condition (73) is amply satisfied inside the solar
system. This gives the justification for neglecting the A-term in the
treatment of all gravitational phenomena connected with the motion
of the planets Furthermore, we see that the line element of the special
theory of relativity, which is experimentally known to hold in every
system of inertia far away from gravitating bodies, appears in the
Einstein model as a consequence of the gravitational field equations for a
homogeneous static distribution of the distant celestial bodies.
For larger parts of the universe, where the condition (73) is not satis-
fied, we get from (72) for the spatial metric tensor and the dynamical
gravitational potentials
rii = r 72/ #2' r2 = r2 ' ^33 = r 2 sin 2 <9
; }, (74)
n* i . r 4 sin 2 '
7 IK = for * ^ *> 7 = bur I = Y^
y, = 0, X = 0. (75)
The spatial line clement
l (76)
1 & *
is real only for r < R, which defines the extension of the physical space
in the Einstein universe. The total volume of the universe is
R TT 27T R IT 27T
r 2 sinedrd6d<t>
000 000
360 EXPERIMENTAL VERIFICATION OF XII, 133
and the greatest distance from the origin is obviously
if i
L ~- 7 - =. R . - /?sin l g -_ . (78)
Further, since the matter is at rest in our frame of reference, we get
on account of (X. 56) for the total proper mass in the universe
M Q - J /3, dV = fi { dV --- 7r 2 /P/l , (79)
and the mean distance of the matter from the origin is of the order
2~ E '
By means of (79) and (68) or (70) we get
l, ~~xl. (80)
477X1
These relations are in accordance with the relations (XI. 49, 50) suggested
by the considerations in 120 on the nature of the centrifugal and Conohs
forces
The spatial geometry defined by (74)-(79) is of the so-called elliptical
kind. However, the line element (76) allows also a somewhat different
interpretation as regards the geometry of the universe as a whole. If
we define four new variables ?/ , y lt ?/ 2 , ?/ 3 by
?/
(81)
?/ 3 r sin 6 sin </>
where ?/o+2/i +#!+#! = ^ 2 , (82)
the hne element (76) takes the Euclidean form
dv* - dyl f dyl+dyl+dyl (83)
This shows that the physical space of the Einstein universe may also be
interpreted as the three-dimensional surface of the sphere (82) of radius
R in a four-dimensional Euclidean space with the Cartesian coordinates
Defining polar coordinates R, J/T, 6, <f> on the sphere (82) by
2/ ~ R cos iff, y l R sin ifj cos 9
(84)
\\e get r ~ Rsiinfj (85)
XII, 133 THE GENERAL THEORY OF RELATIVITY 361
and to each point on the sphere corresponds one set of values of the
variables (0, 9, </>) in the intervals
< l/f ^ 77, ^ < 77, 0<0<27T. (86)
By using (84) in (76) the line element assumes the form
da 2 = # 2 (d^ 2 +sin a d0 2 +sinV sin 2 d<j>*), (87)
and since in these coordinates
y = \y LK = _R 6 sin 4 </fsin 2 0,
we get for the total volume of the universe
7T 7T 7T TT
V = $ dif> j d0 J ^ ^snvtysinfl = 477# 3 J sinV ^ ^ 2^ 3 - (88)
000
Thus the volume in this so-called spherical space is twice the volume
in the corresponding elliptical space. This is also clear from (85) and
(86) which shows that the elliptical space covers only the hemisphere
corresponding to ^ </r ^ |TT or, in other words, antipodal points on
the sphere are counted as one point only in elliptical space.
Similarly we find from (87) that the total distance around the closed
spherical Einstein universe is
L = 2R difj = 277.R, (89)
o
which thus represents the distance one would have to travel along a
'major circle' on the sphere (82) in order to return to the starting-point.
The relations (80) remain true also m the spherical Einstein universe.
Since the dynamical potentials (y t , %) are zero, the gravitational force
on a particle is zero and since, further, the system of reference is rigid,
a free particle in the universe is moving with constant velocity along the
straiyhtest line compatible with the spatial geometry of the universe (see
end of 110) In this generalized form the law of inertia is thus valid
also in the Einstein universe. A free particle at rest will therefore
remain at rest.
According to (X. 94) and (75) the velocity of light is constant and
equal to c. Light emitted by a star at rest at the point (r, 6, </>), at the
time ^ will therefore arrive at the origin r = at the time
362 EXPERIMENTAL VERIFICATION OF XII, 133
and since r x is constant, we get by differentiation
A/ 2 --- A^. (91)
Thus the time-interval A 2 between the arrival of two successive wave
crests at the origin is equal to the interval A^ between their emission.
Further, since x = 0, the time variable t is identical with the time shown
by standard clocks at rest and (91) therefore means that the frequency
of the light as determined by an observer at the origin is equal to the
proper frequency of the light emitted by a star at rest.
Apart from small Doppler effects due to individual random motions
of the distant nebulae we should therefore not expect any systematic
shift of the spectral lines emitted by the nebulae. In the actual universe,
however, the work of Hubble and Humasonf shows a definite red shift
in the light from nebulae, which increases linearly with the distance.
This clearly shows that the Einstein model, in spite of its many satis-
factory features, represents only an approximate description of the
actual universe.
134. The de Sitter universe
Besides the solution corresponding to the condition (61) which leads
to the Einstein model there exists another static homogeneous and
isotropic solution of the general field equations arising from the con-
dition (62), viz. 0c2+ = Q (92)
If we add the equations (59 a) and (596) we get in this case
(ab) f --= K (^c 2 +p)a 2 br =
or ab = constant.
By a trivial change of scale of the time variable this constant can of
course always be made equal to 1, which means that b is the reciprocal
ofa: ab = 1. (93)
Introducing y = I/a as a new variable, (596) can now be written
(yr)' = y'r+y - +l-(
which by integration gives
, , , . , A \
yr = r - ! C :r 3 +constant. (94)
u
f E. Hubble, Proc. Nat. Acad. 15, 168 (1929), E. Hubble and M. L. Humason,
Astrophya. Journ. 74, 43 (1931).
XII, 134 THE GENERAL THEORY OF RELATIVITY 363
Since y is regular for r = the constant on the right-hand side of
(94) must be zero. Thus we get from (93) and (94)
where we have put -^ = - ~ - (96)
./I o
Using (95) in (57) we then get the de Sitter line element in
the form
-
MB*
< 97 >
The components of the spatial metric tensor are again given by (74),
i.e. the spatial geometry in the de Sitter universe is of the same form as
in the Einstein model, at least if A is assumed to be positive. However,
the dynamical potentials are not zero in this case; instead we have
y t = 0, x = ~ a- ( 98 )
Thus a free particle is acted upon by a gravitational force
K = -mgradx = h55' >l (")
proportional to the variable r, which means that the law of inertia does
not hold over large regions of space in the de Sitter world. Only in regions
for which
will the line element (97) again reduce to the line element of the special
theory of relativity and the law of inertia is approximately valid.
The equations of motion of a free particle may now be obtained from
(X. 15) by using the expression for the gravitational force given
by (99). It is, however, more convenient to make use of the possibility
discussed in 113 of eliminating the dynamical potentials by the intro-
duction of a suitable set of space-time coordinates. This is attained by
defining new variables r', 0', <f>', t' by the transformation
(.00,
= e, $ = <f>
364 EXPERIMENTAL VERIFICATION OF XII, 134
As shown independently by Lemaitref and by Robertson,^ this trans-
formation leads to the following form for the line element-
ed 2 == e'W(d/ 2 + r' 2 d0' 2 +r' 2 sin 2 0' df 2 )-c 2 dt' 2 , (101)
which is easily verified by direct calculation. Finally, defining new space -
coordinates x 1 ', y' ', z' connected with r\ 9', <f>' by the usual equations
connecting Cartesian coordinates and polar coordinates in a Euclidean
space, (101) may be written
ds 2 -= e 2d 'W(dx' 2 +dy' 2 +dz' 2 )-c 2 dt' 2 . (102)
The new coordinates #', y' , z', t' can take all values from ~oo to -f<x>-
With these coordinates we have
1 __ 1 _ I
- 722 - 733 -^ - u - ~& - ^33
(103)
Vt = ' X =
and the spatial Christoffel symbols are all zero.
Thus, the time variable t' is the time shown by a standard clock at
rest at any reference point. At any fixed time if the spatial geometry is
Euclidean, x', y' ', z' being Cartesian coordinates apart from the common
factor (WIK. The distance from the origin r' to a point (/,#',(/>'),
as measured by standard measuring-rods, is
l^ p/r/V. (104 )
The velocity of light is constant and equal to c.
, d(J __ c MO^
11 d7'- c (10o)
Further, it is easily seen that the trajectories of light rays are straight lines.
This follows at once from the first equations (VITT 87) for the time track
of a light ray. Taking i -=. i = 1, 2, 3 in these equations, we get
d
x
and by integration -- --= a'e -**'"', (106)
aA
where the a L are constants of integration. Hence
dv'/dt' dy'ldt' dz'/dt'
(107)
f G. E Lomaitre, J. Math and P/??y,9 (MIT), 4, 188 (1925)
J H. P. Robertson, Phil Mag 5, 835 (1928)
XII, 134 THE GENERAL THEORY OF RELATIVITY 365
and the equations of the trajectory of a light ray are linear equations
of the form , , , , , ,
x ^o __ y yp __ z ~ z o (IDS)
a 1 ~~ a 2 a 3
Since the light rays are rectilinear in the (x f , y 1 , z')-space we can obviously
apply the usual tnangulation method to determine the parallax of
celestial bodies and in this way determine the distance (104) by direct
measurements.
Let us now consider the propagation of a light ray along the a:' -axis.
From (105) and (103) we get
or ^-fcce-cr/*. (110)
The motion of a light signal starting off at the point x f = x' Q > at the
time /Q in the direction of the negative #'-axis is thus given by
(111)
and we see that, unless X Q < Re'***' 1 *,
this signal will never reach the origin x' = 0.
Although the world corresponding to the line element (102) is appar-
ently infinite, an observer at the Origin will never be able to obtain any
information about the regions outside the 'horizon' defined by (112).
The greatest distance inside the observable world of an observer placed
at the origin at the time t' Q is thus, by (104) and (112),
r >cto/RJ?e-cta/R T> (1 1 O\
J V J.IL, Jt, V li0 /
i e. it is the same for all times in accordance with the intrinsically static
character of the de Sitter universe. Further, since (102) is invariant
against any displacement of the origin, this consideration holds for any
observer at rest in the system of reference considered here. However,
the position of the horizon will of course be different for the different
'equivalent' observers.
In the present coordinates the gravitational force on a free particle
is zero and the equations of motion are given by (X. 20'):
^=0, (114)
366 EXPERIMENTAL VERIFICATION OF XII, 134
where
p. = mu t
W> :==z n ",
(115)
dx'
Further, since the spatial Christoffel symbols are zero in the present
case, we have * i
which shows that the co variant components of the momentum vector are
constant: ^ = constant (116)
However, the contravariant components are not constant, since we have
P l = y l *P K = e-^ R p t (117)
in accordance with the differential equations (X. 19'). From (i!7) we
now get dx , L
dx'fdt' _ dy'ldt' _ dz'jdt' \
r ~&i~~~ ~ 6 2 " ~ ~~P~
where 6 1 , 6 2 , 6 3 are constants. Thus the orbits of free particles are also
straight lines described by equations of the form
"' "' i/ /!/ f* *r
y ~~y* __ z ~ Z Q n i9\
- 9 V AA7 /
ft 1 6 2 6 3 '
but the velocity of the particle is in general not constant. Only if the
velocity at one time is zero relative to the system of reference con-
sidered here will it remain zero; for from (118) we see that
requires p^ = and dx' L /dt' will then be zero for all times.
Any reference point (x' ,y' ,z') = constant may thus be represented
by a freely falling particle in accordance with the remarks at the end
of 113. Since the system of reference is not rigid, the distance from
the origin of a point with constant values of the spatial coordinates will,
however, depend on the time according to (104). Hence, if we assume
that the nebulae may be treated as free particles at rest in the present
XII, 134 THE GENERAL THEORY OF RELATIVITY 367
system of coordinates, their distance from the origin increases with a
radial velocity jj
dL (s M'ltf r ^ 7 /_ df*\
V ^dt' = R^ r ^R 1 ' (120)
which is proportional to the distance I.
This radial velocity of the nebulae must be expected to give rise to a
shift towards the red of the spectral lines emitted by the nebulae. To
investigate this question we consider light emitted by a particle which
is permanently located at the reference point (r', 6', </>'). According to
(105) and (103) the radial velocity of light is
-'/*. (121)
~
Thus, if t[ and t' 2 are the times for the emission of radiation from the
particle and its reception at the origin r' 0, respectively, we get by
integration
o **
dr' = -c e-*'/* dt' or r f = R(e-^ R -e-^ R ). (122)
On differentiation we get for the time -interval A 2 between the arrival
of two successive wave crests at the origin and the time-interval A^
between their emission the following relation:
A^ - A e^-W'*. (123)
Now, since t' is the time shown by a standard clock at rest and since the
velocity of light is c everywhere, we have for the proper wave-length A
of the emitted light ^ _ * ./
Similarly the wave-length as measured by an observer at the origin will be
A = c A4.
Hence we get from (123)
A = X*e&-*V*. (124)
Further, the distance to the particle at the time of observation of the
light is, according to (104) and (122),
I = f '0VR = R(4*t't-4yR-\Y (125)
From (124) we therefore get
AA A-A I
A A
(126)
Thus, if we assume that the nebulae, apart from small individual veloci-
ties , are at rest in the system of coordinates used here, we get an explanation
368 EXPERIMENTAL VERIFICATION OF XII, 1IU
of the actual red shift observed by Huble and Humason To obtain quanti-
tative agreement the constant R in (126) must have the value
R ~ l-66xl0 27 cm. ~ 1-75 x 10 light years. (127)
This value for the 'radius' of the universe is somewhat smaller than the
value (7 1 ) in the case of the Einstein universe. It is true that this explana-
tion rests essentially on the assumption that the nebulae in the mean are
at rest relative to the system of reference defined by the coordinates
(x 1 ', y',z', t'). This assumption, known as Weyl's hypothesis, f is, however,
very natural and has the attractive feature that all nebulae in this model
are on the same footing so that an observer at any other reference point
would observe the same red shift of the light coming from the nebulae
as the observer at the arbitrarily chosen origin of our system of co-
ordinates.
In the original system of coordinates the motion of the nebulae is
obtained from ( 1 00) by solving the first equation with respect to r. Hence
where r' is a constant for each nebula. Thus we see that the nebulae
according to the Weyl hypothesis are freely falling particles which start
at the point r = at t = oo and end at the antipodal point r = R
for t~> +00.
Finally introducing five variables z^ (z Q ,z v z 2 , z 3 ,z 4 ) by the
equations
, =
(129)
we have z* = ^ 2 , (130)
^-o
and the line element takes the form
* 8 = t (*>) 8 - ( 131 )
/i-O
The space-time continuum of the de Sitter universe may thus be pictured
as the four-dimensional surface of a sphere of radius R in a five-dimen-
sional pseudo-Euclidean space just as the 4-space of the Einstein
t H. Weyl, Phys. ZS. 24, 230 (1923), Phil Mag. 9, 936 (1930).
XII, 134 THE GENERAL THEORY OF RELATIVITY 369
universe, according to (72), (76), and (83), can be pictured as a cylinder
with spherical cross-section and the axis in the direction of the time-axis.
Since the equations (130) and (131) are form -invariant under the group
of five-dimensional orthogonal transformations of the variables (z^),
the line element (102) will be form-invariant under the group L of the
corresponding transformations of the variables x l = (x',y',z',ct f ) con-
nected with (z ,...,z 4 ) by the equations (129). The transformations of
the group L thus connect systems of coordinates (x l ) which are equivalent
in the sense of 121. These transformations play the same role in the
de Sitter universe as the inhomogeneous Lorentz transformations in
the flat space of the special theory of relativity. f
Although the de Sitter model leads to a natural explanation of the
observed red shift of spectral lines, this model can hardly be regarded
as a satisfactory model of the actual universe for the following rea-
son. According to the condition (92) underlying this model, the density
and pressure of the celestial matter which give rise to the line element
(97) satisfy the equation o c 2 + ^ ^ ( 132 )
Since /tc 2 :> 0, (1*32) leads to the assumption that p is negative and very
large Even if we permit the existence of cohesive forces in the ideal
fluid filling our model, a value of p of the order of /tc 2 would, however,
be quite incompatible with the properties of any known material. Hence
(132) can be satisfied only if we take the density to be zero or at least
very much smaller than the mean density of the actual celestial matter.
Hence the de Sitter model corresponds to an empty universe containing
no appreciable amount of matter and radiation, the stars and nebulae
being treated then as a kind of test bodies which do not contribute
essentially to the gravitational field. This point of view is, however, not
in agreement with the basic ideas underlying the general theory of
relativity according to which the centrifugal forces and Ooriolis forces,
for instance, are due to the motion of the distant celestial bodies relative
to the rotating system. While the non-permanent gravitational fields
can be explained along these lines in the Einstein model of the universe,
the empty de Sitter model does not of course afford any basis for such
explanation,, the non-permanent fields being here of the same nature as
the fictitious forces in Newton's theory.
These considerations make it probable that the Einstein model repre-
sents a better approximation to the actual universe than the de Sitter
t See rof , Chap XII, p. 364, and C. Moller, Dan. Mat. Fys, Medd. 18, No. 6, p. 40
(1941)
3595 60
370 THE GENERAL THEORY OF RELATIVITY XII, 134
model despite its failure to explain the systematic red shift of the light
coming from the nebulae. If we want to construct a model in which the
advantages of the two static models of Einstein and de Sitter are com-
bined we must obviously have recourse to non-static models in which
the metric tensor is intrinsically time-dependent. Such models have
been extensively studied in the literature. f These investigations include
models which expand from an originally static state as well as models
which undergo successive expansions and contractions. Our present
knowledge of the actual universe, which only covers a limited region in
space and time, is, however, totally insufficient and thus no unique
choice between the different non-static models is possible.
f See the extensive treatment of these problems in Tolman'b book, Relatwity, Thermo-
dynamics, and Cosmology, Oxford, 1934
APPENDIXES
1. Gauss's theorem
LET V be a finite domain bounded by a closed surface a iri a three-dimensional
Euclidean spaco arid lot/(j: 1 , x. z , ^3) be a given function of the Cartesian coordinates
x \-> 2*2* T 3 inside K. Consider the volume integral ~- dr l dv 2 cfa* 3 over the region V.
J &XL
v
Assuming that the surface o- is convex so that a straight line parallel to the a^-axis
with constant values of x 2 and x 3 intersects the surface a m two points only, this
integral may, by partial integration with respect to the variable x l9 be written
J
(1)
V a,
where the integration on the right-hand side is extended o\ er the projection a
of the sin face a on the (.r 2 o: 3 ) -plane, and/ + arid/" are the values off at the two
points p+ and p~ on a whose projections on the (*t 2 2* 3 )- plane he insido the face
element dv 2 dr 3 of cjj Now let da { , d<j~~ be the surface elements of cr at p+ and p~,
respectively, whose projections on the (.r 2 ^ 3 )-plane are equal to the element dx 2 dx^.
Then, if n+ (nj , w a * , nj ) and n~ (nf , /j,j, n^ ) are unit vectors in the direction
of the outward normals of dcr + and da~~ and if .r^ > .rf , we obviously have
nj da+ nida~-^dc 2 dx 3 (2)
Hcnoe (1) may be written
F
wlieic 1 the integration on the light-hand side is extended over the whole surface
<7, and n l is the u^-component of the outward normal vector n. It is easily seen
that (3) holds also in the case whore tin* boundary surface cr is not convex, in
which case a straight line parallel to the j^-axis may intersect the surface in a
greater but even number of points. Similarly, if g(t l9 x 29 a* 3 ), h(x lt x 2 ,x 3 ) arc
two other given functions of the space coordinates we also have
/'"-/"*
(3')
If, in particular, /, g, and h arc the components a^x), 2 ( x ) a 3( x ) f a vector
field a = (a t ) we get by addition of the throe equations (3), (3')
V
or J diva dV ^- J (a n) da = J a n da, (4)
V o a
where a n is the component of a in the direction of the outward normal. The equa-
tion (4) which enables us to transform the volume integral on the left-hand side
into a surface integral represents G aura's theorem (IV. 188).
359560
372 APPENDIXES
Further, if J tK (x) is a three-dimensional tensor field of rank 2, we get, putting
*u = / ^2 = V> ** = * from ( 3 > and < 3/ )
n K )rfa, (5)
F a
i e. the equation (IV. 192).
2. The transformation equations for the four-current density
According to (V. 4, 4') the transformation equations connecting the four -current
densities 8 l and s^ in two systems of mertia>$ and S' must be .such that the equation
gj-O (1)
is a consequence of the equation
- <n
for all possible charge arid current distributions The transformation must there-
fore bo of the form Q'-ffQ 9 9 9} (2)
S i Jt\ 5 l 5 2' 5 3 5 4/ W
where the functions / t must satisfy the relations
for all variations of the twenty variables s l and .9^ j c)s k /cxi which arc* subjected
to the restriction a /^v
<9 j,t " \*7
Multiplying (4) by a Lagrangiari factor A, which may be a function of the variables
(# t ), and subtracting this equation from (3), we get the equation
which must now hold for arbitrary independent variations of the variables s l and
s k j. By variation of the variables s k> i we get
^,, = A( f )8 t ,. (6)
or, using the orthogonality relations (IV. 14),
|(J = A(,K* (6')
^A;
From (6) we get by differentiation with respect to ,? m
and since the left-hand side is symmetrical in k and m we must have
dX z ax*
*? = *;*<
Taking m I - k in this equation we get
i.e. A must be independent of the variables s % also. Hence we get by integration
of(6 '>
APPENDIXES 373
where the f$ lk and A are constants. If the electric charge density is zero in S, i.e.
s v = 0, it must, also be zero in S'. Hence J3 tk =- and the transformations (2) take
the form tt / \ /o\
*t = ^ik 8 k- W
From (IV. 11), (V. 3), and (8) we now get
^ = A 2 ^s fc ,
or p' 2 (l-w /2 /c 2 ) - A 2 /> 2 (l-ttVc a ). (9)
By the same relativity argument as was used in connexion with the equations
(II. 8, 14) it now follows that the constant A must have the value
A=l, (10)
and (8) then leads to the transformation (V 5), showing that the four-current
density is a four-vector.
3. Plane waves in a homogeneous isotropic substance
In the rest system of a homogeneous isotropic substance with electric and
magnetic constants and JJL and p J -- Maxwell's equations (VII. 31, 32)
may be written diyH _ dlvE 0> (1)
cE = ccurlH, /xH cc-urlE, (2)
D - eE, B ^ /xH. (3)
In the case of a plane wave with wave planes perpendicular to the .r-axis of a
Cartesian system of coordinates (x,y,z) the field vectors are functions of x and t
only. From ( 1 ) and from the ^-component of the vector equation (2) we therefore
get *3r w t _ n
- s f=te =E *- H '-* '
and since constant fields are of no importance in optics we may put
E t = H,~ (4)
The y- and ^-components of the equation (2) give
dE y 8H, 8E Z dH
Differentiating one of the sets of equations (5) or (6) with respect to t and using
the other set we see that each of the functions E v , E z , H v , H z satisfies the wave
equation ? 2 ^0
where w = -~ - (8)
VM
is a constant
The general solution of the wave equation (7) is
whero/j and/ 2 are arbitrary functions. The two terms in this expression represent
plane waves travelling with velocity w in the directions of the positive
374 APPENDIXES
and negative .r-axis, respectively. Thus considering a wave travelling in the
direction of the positive r-axis, we can put
E --= "*/( -*/>), E Z ~- c-*g(t-jt/w), (9)
where/ and # are arbitrary functions of the argument t~xjw. From (6) wo then
get by integration
lf y - -fj.-*g(t-jr/w), H t .- p-*f(t-x/w). (10)
The equations (4), (9), and (10) represent the most general expression for a plane
\va\e moving in the direction of the positive r-axis Introducing three unit
v<1 tors n (1,0,0),
e (1) (0,1,0),
e< 3 > - (0,0,1),
these equations may be written
E *-*/(/ -(x n)ftt')ew-] c-*g(i (x n)/V)e (>) ,
H .-. -n-lg(t- (x n)/M')e (1 M-/A-V('-( x n)e< 2 >,
and in this veetoi fonn they remain tine if the system of (Coordinates i.s rotated
so that n does not lie in the direction of the new r-axis e (1) and e^ are then
aibitiar t \ but constant unit vectors peipeiidicular to eaeli other ancl to the wave
normal n
4. Transformation of the gravitational field variables y LK , y t , x> ^ t/c
by a change of coordinates inside a definite system of reference
The ti.msfoimation (VI II 59)
x'^x'^t*), j'* /(i'), (1)
connecting two dilferent coordinate ,s^si(^nis inside the s.unc system of reference,
is ehnincte) i/ed by the equations
th /l ( i 1 -
i =- TTT -^ ' 4 ^ - M - 0. (2)
C^t' 4 C I 4
Thus \\e get from (VIII. 57)
y[i - *[&iyim a[4<//4 4(if/Ai l J^ii). (2')
f loin u Inch we obtain
Further,
i-e. yi'ic ojajfy^ (4)
Thus y LK transforms like a spatial tensoi and r/<7 2 -- y LK djc'dx* is invariant under
the transformation (1).
The transformation (VIII. 103)
x'< --- ji l , x'* -/(.r 1 ) (5)
APPENDIXES 375
is a special transformation (1) in which only the rate and setting of the coordinate
clocks are changed In this case we have
<--g-S-i.
Using this m (VIII. 55) with i -=.- 4, we get
aj+cx 4 4 ai = 0, c*X 1. (7)
If wo want the gravitational vector potentials y[ to be zeio in the piimrd system
we get fiom (3) the conditions
%}>*- Mi -J 2 x/ c3 ) r^M 1 1 2 x /(l2 ) - ' ()
winch by means of (7) lead to the equations (VIII 109)
*V , ___ ?! ^ n
c^^V(H 2 X /c-)ca 4
Putting <r t = - ^ = -^~ t (11)
\J( ( /ii) ~'<J\\
we get from (2), (2 r ), (6), and (S)
4 -<! *1
It is now easily seen that the opeiator - t -fcr t - is mv aiiant undei the transforma-
tion (5), tor we have
rV l c'*' 1 c'r*- * CI K il c'i 4 '
winch togethci with (12) and (8) gives
_i. K ^ r / r . + <I ,^.. (13 )
A straightforw.ird calculation then, shows that the quantity (VIII. 110) which
can be ui itten
tiansforms according to the simple law
under the transformation (5), i e each component of th(^ spatial tensor OJ IK is
multiplied by the intio b<tweon the rates of tho coordinate clocks in the two
systems.
5. Dual tensors in a three-dimensional space
In a three-dimensional space with a positive definite metric, i e. y -- \y LK \ ^ 0,
the quantities , fi ...
1 *<*A =- vy8 [KA (1)
are the covanant components of an antisymmetric pseudo-tensor if S
three-dimensional Levi-Civita symbol defined at the end of 43. Remembering
376 APPENDIXES
that the determinant |y l *| formed by means of the contravanant components of
the metric tensor is equal to y" 1 , we find that the contra variant components of the
pseudo -tensor (1) are
yU ylK ylA
The covariant components of the axml vector H dual to an antisymmetric tensor
H IK -- H l are now i _ LJX
and the corresponding contruvariaiit components aro
H'-Je*l/ lrt = ^8, >fA ff l , A . (4)
For W, and H' we thus get tlio following explicit expressions
1 (5)
(H*,H*,H*)~^(H n ,H n ,H n ) t
which show that the iclations reciprocal to (3) and (4) are
// = eW A , //, =- c lKA H\ (5')
From two vectors a*, 6 l we can build the antisymmetrical tensor c lK a i b K a K b i
and the corresponding dual axial vector is the vector product
c axb (6)
of the two vectors whose components aro obtained by means of (5) Similarly to
the tensor curl, a = ___ -i corresponds the axial vector curia
dr* dx K
The tensor dual to an antisymmetrical tensor V LK ^ of rank 3 is a pseudo -scalar
F = l e ^ KA = ^-l3 1KA ^A. (7)
If H l ~ ^TT ^wA #KA ls tne vector dual to the tensor H IK . the divergence of H is
dual to the curl of the tensor H iK , for, according to (X. 75), we have
_L d ^y Hi L ^ dH *
Vy d& ~ 2Vy lffA fix 1
1
^}- (8)
Similarly by means of (5') and (2) we get
6. The condition for flat space
A flat space is defined as a space in which it is possible to introduce a system of
coordinates which is geodesic at every point. If this condition is fulfilled we
obviously have D _
^iWm = (1)
APPENDIXES 377
at every point, where -# t jy m is the curvature tensor defined by (IX. 99). Conversely,
we shall now show that ( I ) also represents a sufficient condition for flatness.
A vector field a l (x) is called stationary if it satisfies the equation
at every point where T\ r (x) is the ChnstofTel symbol. The equations (2) represent
a set of differential equations for the functions a i (x) which have solutions only if
the integrabihty conditions
f * Q
l ( }
dx m dx l
are satisfied. The conditions (3) may also be written
-** = o. 0')
The integrabihty conditions are thus fulfilled if (1) holds, and in this case (2) can
be integrated and the solution is uniquely determined if the vector o t (P) at a point
P is given. Since ~ l
da* = ^dx l = -TLa'dx 1 ,
ox 1
the vectors m a stationary vector field are, according to (IX. 62), obtained by
parallel displacements. This is in accordance with (IX. 104), which shows in
the case of a vanishing curvature tensor that the result of a parallel displacement
of a vector is unique and independent of the path along which the displacement is
performed. "
Let us now consider four different solutions a$ {af^, af 2) , af g) , aJ 4) } of the
differential equations (2) determined by four given vectors af fc) (P) at the arbitrary
point P. The vectors a\ k} (P) must be taken to be linearly independent for
instance orthogonal to each other. It can then be shown that we get a trans-
formation x /l jc' l (x k ) which leads to a system of coordinates (x' 1 ) geodesic every-
where if we choose the transformation coefficients
K) = &>(*) (*)
In the first place it is seen that the integrabihty conditions (IX. 14) which also
may be written QW1 Q i
*
are satisfied by the expressions (4). In fact, since the equations (2) hold for each
of the vector fields af k)9 we have
doil (JOLT, dx r datir) . ^
rf = ? & - a? -*= -
which on account of the symmetry of FJ 5 in the lower indices is symmetrical in
k and I.
From the Christoffel formulae (IX. 53) and from (6) we now get
= o.
Thus the Christoffel symbols and therefore also dffyjdx'* vanish at every point
in the system of coordinates (x'*). In this system the components of the metric
tensor are thus constants, i.e. we are dealing with a flat space.
378 APPENDIXES
7. The action principle and the Hamiltonian equations for a
particle in an arbitrary gravitational field
Consider a freely falling particle in an arbitrary external gravitational field
For simplicity, we shall make use of the possibility, mentioned in 113, of intro-
ducing a time-orthogonal system of coordinates (#*) = (X L , cfym which y t = g u = 0.
The connexion between the proper time r of the particle and the coordinate time t
is then given by (VIII. 99), i e.
dr = dt(\ + 2 X / C * ~~ u*/c 2 )*, ( 1 )
where w 2 = y lK u l u* y iK x l x K
is the square of the velocity of the particle, and % = x( x ^ 1S the sca l ar gravita-
tional potential According to (VIII, 85) the time-track of the particle is deter-
mined by the variational principle
-o
and, since the integrand is a homogeneous function of the four variables dx' /dX of
first degree, (2) is equivalent to a variational problem with only three dependent
variables x l and with t as the independent variable. f
Hence the motion of the particle is determined by a vanationul principle
analogous to Hamilton's principle in Newtonian mechanics In fact, multiplying
by a constant factor 'tr^c*, we # ^ ^ 01 a ^ variations 8r l , which vanish for t t
and t t 2 ,
8 L(ji^jc^t)dt - 0, (3)
j a
J L(ji^jc^t
where the Lagrangian L is given by
The corresponding Euler equations
are then analogous to the Lagrangian equations of motion in Newtonian
mechanics for a system with the generalized coordinates x l x l (t) From (4) we
et dL
with m and p t given by (X. 12) Further,
(>L w <-y KA K A dx
~dx' r " ~dx~L U U ~ m ~dr l *
Thus, the Lagrangian equations (5) take the form
-- m - ^ 7 )
at CJL
with d c pjdt given by (X. 13), which shows that the gravitational force is given
t See, for instance, R Courant and D Hilbert, Methoden der Mathematischen Physik I t
Berlin 1924, p. 174
APPENDIXES 379
by (X. 17) Using the general formula (VIII. 98) for dr/dt instead of (VIII. 99) in
(4), one could in the same way find the expression for the gravitational force in a
general system of coordinates, where y t ^ and, in particular, the formula (X. 18).
From the Lagrangian form (5) of the equations of motion we can now pass
over to the canonical Hamiltoman form by the usual procedure. According to
(6), the canonical ly conjugate momenta of the coordinates x l are the covanant
components p t of the momentum of the particle Thus, by means of (4) and
(6), we get for the Hamiltoman function
H -- pi^-L ---- m c 2 (l-f 2x/c 2 )(H-2 x /c 2 ~w 2 /c 2 )-i
= mc 2 (l + 2 x /c 2 ), (8)
which is identical with the expression (X. 22) for the energy of a particle in a
gravitational field with y t = 0. In a static gravitational field, H is a constant
of the motion. Solving (6) with respect to w t , we get
which allows us to express H as a function of the canonical variables x l and p t .
We got l '* 2 )*. (10)
It is now easily verified that the equations (6) and (7) are equivalent to the
canonical Hamiltoman equations
<-% *-"
and the whole Hamilton Jaeobi thecny of integration can then also be applied
in this case
In the variation principle (3), the actual motion l - r l (t) is compared with a
virtual motion m which the particle has the coordinates Jc L .**(<) -j-8.r l ($) at the
time /, i e the symbol 8 refers to a variation when; the time is kept constant. We
shall now consider a more general variation by which the time is varied also, so that
the particle in the* varied motion has the coordinates l = #'(0 + A# l at the time
t-{ A/, where A may be regarded as an arbitrary infinitesimal function of t.
Neglecting terms of order higher than the first, we obviously have
A.r l =- 8.ri+j/At (12)
and, consequently,
^
A v*t _ __ A _ _ __ __ _____ __ _ _ _ _
dt ~~ dt ~ ~dt l "~ dt dt
d&i 1 . d&t d&t
_-_ +J -A,^.__^_,
01 Ar l Sx l + x l &t. (13)
Thus, \ve get foi any function L(x L 9 jc l , t) of the variables x l 9 Jt l , and t
Hence, ^(L dt) - &Ldt + Ldkt $L dt-\ -^-dt (15)
(16)
and A f Ldt = 8 fi cft + ( At)
380 APPENDIXES
According to (3) and (16), we thus have
**
A J Ldt^Q (17)
provided that we choose
A* ^ for t = ^ and t = t 2 . (18)
On account of (8) this may also be written
A ^ p^d^-b J Hdt --=- 0. (19)
Now we have
r dt) - //A rf/ |- AH dt ~ H d&t-\- l8H-{-~j- At] dt
(20)
dt
where we have made use of (14) applied to the function H.
Since the A -variation is more general than the 8 -variation, we may now impose
the further condition s u- A / o i \
OJnL = U, ("*)
which means that the energy of the particle at any time has the same value in
the actual and the virtual motion. This does not necessarily mean that the energy
must be constant along the path. From (20) arid (21) we then get
"=0
and (19) reduces to A J p, dr l - (22)
'i
(22) together with (21) and (18) correspond to Maupertuis's principle of least
action in its most general form
For a static field, the energy H is equal to a constant K> along the actual path,
<?
and Maupertuis's principle states that the 'action' J p L dx l m first approximation
*i
has the same value for all virtual motions with the same energy E. In the static
case, the time may even be eliminated entirely from (22). Since p L = mw t and
dcr/dt u> we have
p t dx l p t u l dt - mil* da/u - - mu dcr =- p da
and (22) may be written (&)
0, (23)
where (x\) and (jc) & T this coordinates of the particle at the times t t and t 2t
respectively.
Putting H K in (10), and solving with respect to p, we get
p = {(J5/ C p-m; C 2 -2mi; x }i(l-f2 x /c2)-J. (24)
Henco, the variational principle takes the form
Ui)
A (l + 2 x /c 2 )-4{(^/c) 2 ~m?c 2 -2^ x }*d<T - 0, (25)
APPENDIXES
381
where E as well as the end points (x\) and (x^) are to be kept constant by the
A-vanation. In this form, the variation principle is analogous to Hertz's 'principle
of the straightest path' in Newtonian mechanics. In the limit of weak fields and
small velocities the integrand in (25) reduces to
V{2m (e-w x)}, where -= E-Jk Q c 2 ,
which is the integrand occurring in Hertz's principle.
If ^ = 0, the integrand in (25) is a constant and we get
8 J da - 0,
(26)
which shows that the path is a geodesic in the space defined by the metric tensor y t/c .
As remarked by Hertz, f the more general case, where x(x i ) = 0, can formally
be treated also as a problem of determining a straightest path, but in a space
where the line element is defined by
dS - T IK dx l dx K = p 2 da 2 >
(27)
Such a treatment is, however, quite formal, since the geometrical structure of the
space as determined by natural measuring sticks is described by y tK and not by F tK .
8. The connexion between the determinants of the space-time
metric tensor and the spatial metric tensor
According to (VIII. 64, 63) we have
044
(1)
The determinant y may be written in the form
0i4
024
o o
o
Since a determinant is unchanged if we multiply the elements of one of the
columns by a common factor and add them to the corresponding elements of
another column we have
,014014
yi2 yis 0i4
0u yi2 >
'13 014
044
, ,024014
y2iH ~
044
y22 y23 024
021 y22 >
'23 024
^ ,034014
ysiH -
044
y32 y33 034
031 ys2 y
'33 034
044 014
g u
041
044
044
on account of (1).
t H. Hertz, Die Prinzipien der Mechamk, 2nd ed., Leipzig (1910).
382
APPENDIXES
If we apply the same procedure to the second and the third column we get
011 012 013 014
021 022 023 024 / \
W
031 032 033 034
041 042 043 044
Since both # 44 and g are negative, y must thus be positive and we have
V("0) = \ / (-044> Vy Vy V(l+2 x /c 2 ). (3)
9. The derivatives of the function with respect to ffi and g lm
and some identities containing these derivatives
According to (XI 132) the function i* can be written
where A - g lh \\Y s is < B = g lr rf m Y L (2)
Hence, since </'* and g tk are kept constant in partial derivation Mith respect to
</j! m we have
*-'"-- ~- * ^^^ .
61,4
From (IX 68') we got
Further, by contraction of (IX. 68) with respect to the indices k and
Hence
(6)
(7)
(8)
winch is symrnctrji al in / and ?// (cf. tlie remarks following (XI 140)).
Thus, by means of (4)- (8), w(^ get
8 A
In order to find the corresponding dei i\ ativc of B we consider a variation of the
variables gff 1 for constant g lk The corirspondmg variation of B is
87* ^dLSiirsr
If, in the last term, we perform a cyclic permutation of the four summation indices
k, I, r, m we get &B ___ rf m S(g l 'rf k \ gT\ k ),
which by means of (IX 68) reduces to
Hence
SB
(10)
APPENDIXES 383
which together with (9) and (3) leads to the desired formula
*~)O
^ - V(-flO{r{m-i(8? r^+s^rf^-K^Tfc-flf^r;.)^}. (ii)
In the same way we can find the derivative of i* with respect to y l>H . However,
it is easier to use the equation (XI 139) according to which
The last term may be obtained from (11) by differentiation with respect to x k
while the first term is given by (IX. 114, 111) Besides the components of the metric
tensor and Christoffel symbols these expressions contain derivatives of the
Christoffel symbols. Since fl and therefore also the left-hand side of (12) does not
contain derivatives of Christoffel symbols, all terms on the right-hand .side con-
taining such derivatives must cancel arid we may therefore omit them from the
beginning in our calculation. By differentiation of (11) with respect to .*,* we are
then left with terms containing derivatives of ^J( g}, g, gi m , only By means of
(IX 69', 68, 51) these terms and therefore also the right-hand side of (12) may
be expressed in terms of components of the metric tensorand of Christoffel symbols
After a straightforward calculation one finds
0,1, - V(- sOl-i'&rmr-rfrrj^- i(0j* -j^r^Mi^H- i\ mk )}- ^ m . (is)
The expressions (11) and (13) for the derivatives of with respect to g l and g lm
are easily seen to be in accordance with the equations (XI 141, 142).
It we now substitute (11) and (13) m the last term $]{ on the right-hand side of
(XI. 101) and use the equation
following from (IX 68), we get for * an expression containing a large number of
teims, which, howevei, cancel in pairs. The final result is therefore that the
quantities i*J are identically zero
1 . d 82
~~ -
AUTHOR INDEX
Abraham, M., 88, 160, 193,
204, 205.
Adams, W , 348.
Airy, G -B , 26
Anderson, C D , 91.
Bambridge, K T , 90
Beauregard, O C. de, 170.
Becker, R , 160, 204
Belmfante, F. J , 186
Bergmann, P. G., 286.
Bethe, H., 90.
Blackett, P M S , 91.
Bohr, N., 346.
Born, M , 75, 157, 194, 195.
Bradley, J., 25.
de Broghe, L , 58, 105
Bucherer, A. H , 89
Campbell, W. W., 355.
Champion, F. C , 85, 88
Chazy, J , 352.
Cockcroft, J. D , 89.
Courant, R , 378.
Curie, J , 91.
Dallenbach, W , 195, 204,
206
Dirac, P. A M , 91, 98.
Einstein, A., 30, 31, 32, 41,
46, 49, 82, 139, 211, 219,
258, 310, 313, 321, 327,
338, 348, 355, 356, 358
Eotvos, R v., 220.
FitzGeralcl, G. F., 28.
Fizeau, H , 10, 19, 20
Fokker, A. D , 170, 195.
Foucault, L., 10
Fresnel, A.-J., 16
Freundhch, E., 348.
Fried lander, B., 320
Fnedl&nder, T., 320.
Galileo, G , 220.
Gerlach, W , 89.
Guye, Ch. E , 89.
Hasenohrl, F., 211.
Heisenberg, W., 195.
Herglotz, G., 179.
Hermann, W , 10.
Hertz, H , 381.
Hilbert, D, 311, 314, 378.
Hoek, M , 17.
Hubble, E , 358, 362, 368.
Humason, M L., 362, 368
Huyghens, C , 1, 15
Ilhngworth, K K , 28
Ives, H E , 10, 62
John, S , 348
Johot, F , 91.
Jordan, E. B , 90.
Kant, I , 224.
Kaufmann, W , 89.
Kennedy, R J , 28
Kmoshita, S , 10
Klein, F , 338
Kohlrausch, R , 5
Langevin, P , 85, 258
Laue, M. v , 175, 204, 258
Lavanchy, C , 89
Ledermann, W , 348.
Lemaitre, G E , 327, 364,
369
Lense, J , 317
Levi-Civita, T , 309, 333.
Lewis, G N , 67
Livingston, M S , 90
Lodge, O., 28
Lorentz, H. A , 21, 22, 24,
28, 29,30,40,46, 82, 193,
195, 258
Maxwell, J C , 6.
Meitner, L , 91
Michelson, A. A., 15, 19,24,
26, 28.
Mie, G., 194.
Miller, D. C , 28.
Mmkowski, H , 93, 105, 136,
139, 149, 160, 195, 203
Moller, C., 170, 190, 258,
299, 369.
Morley, E. W., 19, 26
Neddermeyer, S H , 91
Neumann, C , 356
Neumann, G , 89.
Newton, I, 4, 219.
Ocehiahni, G P. S., 91.
Papapetrou, A , 170
Pauh, W., 204, 327.
Phihpp, K , 91.
Planck, M., 181, 211.
Pomcare, H , 93, 139, 194
Pryce, M H L , 170
Rasetti, F., 91
Reissner, H., 333
Robertson, H P , 364, 369
Rosonfeld, L., 21, 186.
Sagnac, G , 64
Scheye, A , 209.
Schwarzschild, K , 326,
330.
Seehger, H v , 356
Serim, R , 327
Siege!, K , 10
Sitter, W de, 317, 356.
Smith, N M , jr , 90
Sommerfeld, A , 75, 129,
144
Southerns, L , 220
Stark, J , 10
Stilwell, G R , 10, 62
Stokes, G G , 15
Synge, J L , 327
Tamm, J , 204, 205, 206.
Thirrmg, H , 317, 320.
Thomas, L W., 56
Tolman, C., 67, 204, 214.
341, 357, 370.
Trumpler, R , 355.
Walton, G T. S , 89.
Weber, W , 5.
Weyl,H., 157, 309, 333, 368.
Weyssenhoff, J. v., 250.
Yukawa, H , 184.
Zeeman, P., 63, 220, 221.
SUBJECT INDEX
Aberration of light, 25, 62.
Action principle for a particle in a gravita-
tional field, 378, 380.
Addition of velocities, 3, 52.
Affine tensors, 273,
Angular momentum, 110, 138, 169, 189.
Bianchi identities, 286.
Black -body radiation, 216
Centrifugal force, 4, 218, 317.
Chris toff el formulae, 273
three-index symbols, 273
Clock, coordinate, 226, 235
paradox, 49, 258
rate of moving, 48, 247
standard, 33
Closed system, centie of mass of, 170,
definition of, 163.
Conservation of electric charge, 140, 197,
302
of momentum and energy, 163, 337.
Coordinates, Cartesian, 92, 231.
curvilinear, 228, 233
equivalent systems of, 321
Gaussian, 228.
geodesic, 274
pheudo-Cartosian, 233
quasi -Gali lean, 342
time -orthogonal systems of, 238, 296
Conohs force, 4, 218, 317
Cosmological models, 356
Covariance of the laws of nature, 97, 265
Co van ant differentiation, 280
Curl, 126, 127, 283, 284
Cuivature tensor, 284, contracted forms
of, 286
Deflexion of light in a gravitational held,
353
de Sitter universe, 362.
Divergence of a tensor, 127, 283
of a vector, 127, 283.
Doppler effect in de Sitter universe, 367
non-relativistic, 8-10.
relativistic, 62.
Dual tensor, 114, 270.
Dynamics of a particle in a gravitational
field, 258, 290, 295, m the special
theoiy, 105.
Einstein universe, 357.
Elastic matter, 173.
stress, momentum density, and
energy density of, 175-81.
Electrodynamics in a gravitational field,
302, in stationary matter, 195, in uni-
formly moving bodies, 196, in vacuo,
139.
Electromagnetic field tensor, 141, 196,
302.
Electrons, classical model of, 193, theory
of, 20.
Energy, conservation of, 163, 337.
gravitational, 340.
kinetic, 70.
of a particle in a stationary. gravita-
tional field, 294.
transformations of, 28.
Energy-momentum tensor, Abraham's,
204
elastic, 176.
- electromagnetic, 159, 307.
for general fields, 185.
kinetic, 136.
Mmkowski's, 202
for perfect fluids, 182, 300.
total, 161, 163.
Equivalence of energy and mass, 78;
principle of, 220, 264.
Equivalent systems of coordinates, 321.
Kuler equation, 184, 229.
Format's principle, 23, 308.
Fizeau's experiment, 19
Flat space, condition for, 376.
Force, electromagnetic, 155, 156, 203, 205,
306.
fictitious, 4, 218, 219
gravitational, 291
transformation of, 70, 73
Four-acceleration, 102
current density, 140, 141, 197, 302
force, 105, 295
momentum, 104, 289
ray velocity, 103
vector, 99, 266
velocity, 102, 288.
wave number vector, 103
Fresnel's dragging coefficient, 16, 63.
Galilean transformation, 2, 250.
Gauge transformation, 144, 248.
Gauss's theorem, 128, 371.
Geodesic lines, 228, 272
system of coordinates, 274.
Geometry, non-Euclidean, 226.
Gradient of a scalar, 126, 279.
Gravitational field, static, 250, 323;
stationary, 250, 294.
386
SUBJECT INDEX
Gravitational Held equations, 310, linear
approximation of, 313
Gravitational mass, density of, 344
shift of spectral lines, 346
Hamiltoman equations for a paitiele in un
external gravitational field, 379
Hook's experiment, 1 7
Huygheris principle, 11.
Hyperbolic motion, 75
Ideal monatomic gases, 215
Incoherent matter, 130, enorgy-momen-
tum tensor for, 1 36, 300
Inert lal system, I.
Interval, 99.
Kronocker symbol, 94
Levi-Ci vita symbol, 113
Local systems of inertia, 1 04
Lorentz contraction, 28, 44, 96
transformations, general, 41, 92.
infinitesimal, 117
special, 36, 95.
successive, 53, 118; without rota-
tion, 42, 118.
Mans of a cloned system, 77
of a material paitic lo in a giavitational
field, 290, in a system of meitia, 69
Meson fields, 184
Metric tensor, 228,
experimental determination of, 231,
237.
properties of the space-time, 235
space-time, 233
spatial, 238
Michelson's experiment, 26
Moment of force, 1 1 1, 138, 190
Momentum of a material particle in a
gravitational field, 290, in a system of
inertia, 69.
transformation of, 71
Non-closed system, definition of, 188
Parallel displacement of \octois, 276
Particle velocity, tiansforination of, 3,
51, 52, 53
Perfect fluids, 181
Perihelion, advance of, 348
Permanent gravitational fields, 221
Phase velocity, ttansfoii nation of, 8, 23
Potentials, electromagnetic, 143
dynamical gravitational, 246
Potentials, elimination of gravitational
dynamical, 296-8
Li&nard-Wiechert's, 149
retarded, 148, 315
Poynting's vector,
Pseudo-tensor, 112, 270
Rate of moving clock in a giavitational
field, 247.
Hay velocity, transfoirnation of, 11, 15,
58
Reference points, 234
systems, ^eo systems- of reference.
Relativity of centrifugal forces and
Conolis forces, 317.
general pimciple of, 218
principle of mechanics, 1-4.
-- special piituiple of, 4
Retardation of moving clocks in a system
of inertia, 48, 49, 97
Schwarzschild's oxtonor solution, 325,
anterior solution, 328
Simultaneity of events, 31, 33
Static giavitational holds with spherical
symmetry, 323
- non-closed systems, 191
Systems with sphencal symmetry, 322
of refeience, general accelerated, 233,
234
inertial, 1, 17
rigid, 253
unifoirnly lotating, 222, 240
Tensor, 108, 111, 269
ami psoudo -tensor fields, 125, 279
Thermodynamics, foui -dimensional for-
mulation of, 214
in stationary matter, 21 1 , in uniformly
moving matter, 212
Thomas precession, 56, 121, 125
Time-oithogonal system of coordinates,
238, 296
Tune track office particles and light rays,
244
Vanational pimciple of electrodynamics,
157, for geodesies, 299, for gravita-
tional fields, 333 , for time tracks of free
paiticles and light rays, 244
Velocity of light in gravitational fields,
240, 308, in refractive media, 15, in
7'arwo, 10
Velocity of propagation of the energy, 164 ,
ma lightwave, 161, 206
Work, 70
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